Your culture can survive on these staples, however for what we call complete nutrition today, you would do well to add a leafy green, a root vegetable other than the potato, a cultivated berry (for a more reliable and plentiful supply than foraging), an ungulate like a goat, camel, or cow, and some fish or shellfish.
Your listed foods are going to cover the macronutrients (major sources of calories; carbs, protein and fat) with no problem. Most historical cultures only had one staple carbohydrate source, so to have two makes you that much more protected from famine. The anachronism and cultural difficulties of cultivating rice and potatoes have been covered by other answers, suffice to say, consider wheat instead of rice or yams instead of potatoes.
There are 13 essential vitamins, of which you're only going to have trouble with three. The egg yolks are going to provide you with most of these vitamins, while grains will round out the majority of B-complex vitamins, and you'll get a vitamin here or there across the rest of your listed foods.
Vitamin B9 (folic acid) and Vitamin K are normally produced in ample amounts by the human body, but a little extra of either never hurt, especially for pregnant women and young children. However, the only listed food they'll be found in are the livers of your chickens; one or maybe two chickens will feed quite a large family, but the livers would only be big enough to give each family member a small bite, requiring you to prioritize. Consider a leafy green like a lettuce variety, arugula, mustard greens, spinach or cabbage, depending on average daytime temperature. All of these contain both of these vitamins in abundance and don't require killing too many of your egg-layers just for liver.
That liver is also your best listed source of Vitamin A, though egg yolks have quite a bit of both Vitamin A and of "carotenoid" precursors that can be converted into the vitamin. A red to orange root vegetable like carrots, beets or yams have good carotenoid content, as does the spinach already mentioned, and various gourd-family fruits like zucchini, butternut squash and pumpkin.
The third is Vitamin C, which is found in most species of edible berry, including currants, blackberries, raspberries and blueberries among others, all of which are native to most of the northern hemisphere. Consider cultivating these for a more plentiful supply; as native vines and bushes, they're not hard to grow where you want them, don't require much attention, and can thrive in rockier areas of your farmland that aren't suitable for other crops, or as hedgerows between farmed plots.
There are 16 essential minerals in modern nutrition, most of which you are going to have no trouble with between the chicken and the beans. One major one you're going to miss with what you have is soluble calcium. The most common source today is in milk, and you have no milk producer in your list. Legumes and leafy greens give you a little calcium, but it's a mineral that vegetarians and especially vegans historically struggled with until soybean products became more widespread in Western agriculture and cuisine. A herd of goats don't take much room, can eat a variety of vegetable matter including grass, and produce a milk very rich in calcium and other vitamins and minerals. In addition to the milk, bone meal, made simply by drying and grinding the bones and gizzards of the animals you slaughter for meat, has historically been an important source of calcium, both dietary and as fertilizer (which gets calcium into your crops and thus into your diet).
Also keep in mind that the goats (and other mammals) have to give birth regularly to keep lactating, and that only female goats are really useful for this purpose beyond a couple of rams to keep the female goats pregnant. So, much like your chickens, farmers artificially keep a very female-heavy population by culling the males for meat, giving you a second good protein source (and a red meat like mutton is more nutrient-dense than chicken). The goat skins are also a key source of leather, which is critical to medieval technology above and beyond your nutritional needs. Sheep are another good option especially in colder weather, as you not only get milk, meat, skins etc but also wool (there are hairier breeds of goat, but mohair is a relative niche fiber compared to wool).
Another key mineral source you may struggle with in a more inland area is salt, which provides the key electrolyte ions sodium and chlorine. This is normally trivial to get in abundance for any culture with access to the ocean, since all you have to do is boil clean seawater dry and presto, sea salt. However, it can be more of a challenge to get sufficient salt intake in more inland areas. The saltier the soil, the less fertile it is, and while there are salt mines dating back to 5000 BC, these aren't nice places to work for a variety of reasons (historically they were mined by slaves or prisoners with extremely short life expectancies) and aren't found everywhere humanity has settled, so a landlocked haven for a medieval European farming culture is going to produce ample but very low-sodium produce, and will have to trade for its salt. The East Asian answer is soybeans, but these didn't spread out of Asia as a food crop until the late 1700s, and most of their sodium content as a foodstuff comes from boiling them in seawater to extract and neutralize their trypsin inhibitors; boiling anything else in that much salt is going to put plenty of it in the food. You can get enough of these electrolytes with a meat-rich diet, or by drinking a sparing amount of animal blood (blood plasma is by definition an isotonic solution of electrolytes, but human digestive systems don't process whole blood very well), but if chicken is your only animal protein it will be a bit harder to manage. The easiest and least gory answer all around is to make sure you give your fictional community good access to a body of salt water.
The last one you could have some trouble with is iodine. Essential for proper thyroid function, in turn regulating growth and metabolism, you can get enough of it as long as the soil you grow your staples and vegetables in is iodine-rich. Geographical regions to avoid on Earth include sub-Saharan Africa and heavily mountainous regions; almost anywhere else on the planet, especially in coastal wetlands, you'll get sufficient iodine from vegetables and grains grown in the local soil, but fish or shellfish like mussels would be an ideal addition to your group's diet for a little extra.
