I'm trying to make an empire that has a large (almost as big as the Sahara) desert in the middle of its boundaries. The capital would be found in the north near a mountain range, and the empire follows the continent's shoreline on either side of the desert that both meet a whole southern part. Basically, a donut with a desert in the middle. Could an empire cope with geography like that? Could they just work around the desert?
Edit: Pre-Industrial Revolution Era
perfectly possible. We have examples in our own history.
The Mongol Empire had a similar layout, around the Gobi desert and the Asian steppes
The Mongol Empire of the 13th and 14th centuries was the largest contiguous land empire in history. Originating in Mongolia in East Asia, the Mongol Empire at its height stretched from the Sea of Japan to parts of Eastern Europe, extending northward into parts of the Arctic; eastward and southward into the Indian subcontinent, Mainland Southeast Asia and the Iranian Plateau; and westward as far as the Levant and the Carpathian Mountains.
Also some Caliphates started around a desert.
If your main concern is that the empire would be "thinned out" by the massive desert then there's something approximating a historical counter-example. The situation you describe is very analogous to that of the Roman empire at its height where the Roman "desert" was the Mediterranean Sea. The big difference, of course, is the Mediterranean did allow pretty easy trade, troop movement, etcetera, by ship. Clearly you can't build settlements on the ocean and unless you're a fisherman on the coast the resources provided are pretty negligible, so the situations match in those respects. I'd suggest that as long as caravans can cross the desert for the purposes of trade and occasionally troop movement then your scenario is entirely plausible.
Edit: As an aside, if the empire can build in the desert I could see a scenario where it might actually put its capital right in the middle. As long as that city has a supply network by which it can get food and water from the perimeter and good stores for each it would be very hard to besiege such a city. While under attack, the capital could pursue a scorched earth policy with any outposts and basically just hold up while the enemies water stores run out faster than the city's. This could create a really interesting dynamic where the main city/cities are lightly defended and the military interests of the empire are almost entirely focused on sustaining and controlling supply chains since its urban core is not self-sustaining. In a more economic sense, Singapore is a little like this. With so few natural resources it imports a ton of basic essentials including (if memory serves) the majority of its water, but its population is highly educated and productive in other ways that fund this. You could have a desert-Singapore/Rome hybrid. Just an idea to take or leave.
Edit2: Keep in mind that the Roman military was as famous for its engineering feats as it was its combat feats. When Caesar warred against the Gauls, there were apparently cases where his armies would build fortifications outside enemy cities literally overnight which were superior to the cities fortifications. Shifting to the civilian side, the empire was also covered in roads of such high quality that some are still used today. Then there's the aqueducts... your hypothetical civilization is very likely one that prides itself of infrastructure, engineering, and logistics. Being good at these things mitigates the disadvantages of not have large amounts of contiguous convex area while leaning into the advantages of having great reach.
You are worried that the empire will lack transport and resources to allow it to function as a unified whole and have large-scale production. I think the key factor you need to have is cultural unity. Your empire will have a culture and society centered around the desert. THAT will be the factor that holds your empire together.
First, let's establish that it is the non-'high'-desert areas in which your populace lives. It is the minimum amounts of shepherding, agriculture, and such that allow people to live in sustainable numbers. It doesn't NEED to be easy - in fact, overcoming challenges via organization and logistics is often what moves a culture from a kingdom to an empire.
Your empire needs unifying factors that keep otherwise tough individualist desert nomads working together with soft bureaucrats and farmers. Religion is an excellent motivator, and if you have a faith emphasizing each group's critical role in society, this can tie it all together.
Your empire needs a source of wealth. Trade is the logical one, as the desert can have critical trade routes going through it. Gold hidden in the desert (mines) is also a good source of wealth. Gold can lubricate trade, and they go well together.
So a culturally unified desert society with a central faith and wealth through trade, where the neighbors are tolerated but considered outsiders, would be difficult to invade (centered as it is around a desert) and resistant to civil war. If survival depended on the logistics of the empire, people would work hard to maintain internal cohesion to preserve the whole.
YES with faith in Government
If a nation has faith in their government, then distances or hurdles do not matter. Whether it is desert, mountain, ocean or even some enemy country in between, the nation remains united.
In Pre-Industrial Revolution Era, there are many examples of such states.
