First of all, although it might be possible for a single empire to control the whole or most of the planet, it is more likely that there will be at least 2 superpowers, together controlling most or all of the world, but being separate and either opposing eachother or being neutral until an opportunity arrives. Realistically, you would have 3-6 empires, though alliances would make them feel like 2-3. So even if you had a single empire, it is likely that sooner or later it will break up into a few pieces. As long as the pieces are allied or neutral, all is good, but if the pieces are active enemies, there is a chance for further splitting into smaller parts. Now, if the empire is somehow spanning multiple planets, like if portals/wormholes are present to connect the two, or the gravity is weak enough for space-towers to connect the two, then it might be easier to have a planet-wide empire.
This is the bare minimum needed for an empire to control the whole planet: Optical telegraph, trains, mirror-based telescope, masts, airplane wings, fans (of any kind) and hot air balloons (to mix them to get an airship), and reduction gears. Extra points for steam engines, stirling engines and windmills, waterwheels and aerial ropeways.
- The optical telegraphs are used to send and receive messages.
- Mirror-based telescopes are used to check on other nearby settlements, to see if -something is wrong, and for mapping.
- Airships powered by pedals or treadles for mapping and traveling without having to use roads, or to change from carts to ships and back. Reduction gears would be used for moving the fans at the wanted speed.
- Trains powered by pedals/treadles/hamster_wheels and using human or animal power to move, and aerial ropeways to help it move uphill or accelerate faster when leaving the station. Reduction gears would be used for moving the wheels at the wanted speed.
- Steam engines for ships and airships (not necessary airplanes), for when the masts are not delivering enough power, or when you would rather hurry.
- Stirling engines are for mechanical power generation, and for refrigeration if you run them in reverse, feeding them movement and moving heat instead of feeding them temperature difference and receiving movement.
Using tall towers for the optical telegraphs, you can increase the distance you can talk with both fixed and moving targets. Usually, the fixed targets will be talked with by using strong lights and mirrors to blink/flash towards the target, and the target will use a telescope to look at you and see the signal. For that reason, it is likely that different towers or different coordinates will be used, often with some space nearby to make sure the target is not thinking you sent a message when in fact it was a different tower.
The optical telegraphs would be first used for mapping, and next would be using airships for mapping. Due to the danger of airborne attacks, the medieval castles will evolve to oversized ziggurat cities, with mirrors sending a lot of sunlight to the underground, since using torches would be rather dangerous.
With airships, you can get a better picture of the continents, also useful for maps, and also travel for long distances without worrying about mountains and water currents, but instead worrying about wind currents.
We had bearings, both plain and roller and ball bearings since around the time Roma was born. There were also trains used for a short time period around that time, too. Even without steam and stirling engines, just with wind power, water power and pedaling, we can have an empire span the whole globe.
The trains would be used for moving things on a long distance using windmills, waterwheels and pedals/treadles/hamster_wheels. Plus that trains would be used to move large quantities of troops and resources to deal with rebellions and other issues. If need be, a pedal-powered fast blimp might be used to deploy some shock troops, or elites, to deal with the problem immediately.
Stirling engines can be used instead of steam engines and watermills and windmills for a distributed mechanical power generation network, probably using mechanical tension on ropes to move the power around. Alternative ways to move power would be compressed air, compressed water (tap water) from a water tower (or more water towers) used as storage and compression chambers. The energy can be supplemented with wind power and/or water power, if needed or wanted.
That's it. That's all one would need. And that is still a bit much.
Edit: You can use pedal/treadle powered vehicles to move faster with either human power or animal power. Just get an animal or human on an oversized hamster wheel (like those thread wheels used in medieval times to move heavy things up in elevation), and you can use their output to achieve high speeds. With or without rails. During the US's Great Depression, people tried using pedal-powered cars, but shortly after that there was a lowering in fuel price, which caused the idea to be abandoned by most people.
There is a lot more which can be done with that, so ask me if you want to know more.
Edit 2: It would also be useful to have a technological advantage over the rest of the world, to be able to give important gifts to the taken over people, to win them over and ensure rebellions will not happen. Since papyrus can be made, and wax and resin were used to waterproof things, there can be an empire which uses waterproofed papyrus paper boards for greenhouses, which might be cheaper and require lower tech than glass does, plus that you would use glass for the mirrors needed for the telescopes for the optical telegraph. Greenhouses will also keep your existing people fed and give them cheaper condiments. For growing temperate plants in the desert, you would likely need some way to cool the air, like through pipes going underground for tens of meters, before getting into the greenhouses where the amount of light is limited by having a few small windows allowing a lot less sunlight inside, which paired with the colder air can sustain temperate-climate plants. The humidity would still be an issue, though. Being able to get spices and grow temperate plants in colder and hotter climates surely does make up for that.