How to build a habitable desert planet, maybe. Mostly speculation, since I'm far from being an expert*.
The fundamental problem here is that in order to have a breathable-by-humans atmosphere, you need photosynthetic life to generate atmospheric oxygen. (Since oxygen is highly reactive, it needs to be continually regenerated, too. See "carbon cycle".) Life as we know it needs water to exist. Even the most highly desert-adapted plants need water, and indeed, many of their adaptations are for obtaining and conserving the limited water in their environment.
Furthermore, in order for life to originate in the first place, you probably need an ocean environment. It's also arguable that you need some oceans to act as a CO2 sink over geologic time (limestone & chalk), and as a regulator in the shorter term. So your planet must have some ocean, or at least have had them in the past.
Another reason you need water is for plate tectonics to happen, which might well be necessary for habitability - think of volcanos adding all sorts of stuff to the atmosphere, and the ocean crust being continually recycled by subduction.
The bottom line here is that you're just not going to get a habitable planet that's entirely desert. You'll have to settle for one that's mostly desert.
So you have a planet that's midway between Mars & Earth, say 70-80% of Earth's mass, and similar composition, with a molten iron core to give you a protective magnetosphere. It either started out with less water than Earth, and/or lost much of it over the eons. It has active plate tectonics, and so mostly granitic continents and basaltic ocean beds. Oceans exist, but they occupy only trenches and the lower parts of what would be oceans** on Earth.
You still have evaporation from the oceans, and so rainfall and all the other weather that Earth has. However, orography ensures that almost all
of the precipitation falls on the continental slopes (the area between the continental shelf and the deep ocean). Very little reaches the continents.
So you have Earthlike continents that are desert, rimmed by wetter areas. Your humans naturally inhabit the wetter areas (where they can grow food), but venture into the deserts for trade, mining, &c. A Columbus on your world would use camels, not ships.
Now about those ice caps. On present-day Earth, they're limited by the fact that ice flows, and when it reaches the ocean, chunks break off and float equator-wards, where they melt. On your desert planet, the oceans are much lower, so this won't happen. Instead, the ice will be more like that of Ice Age North America, flowing towards the equator until it reaches a balance with melting. Thus your continents might have a number of meltwater rivers flowing through the desert. (Like the Nile.) There might even be occasional lakes that would be the equivalent of islands in the ocean.
*Here's a book by someone who is: https://www.cambridge.org/core/books/principles-of-planetary-climate/5B5EEF0534CB6F69FB2E395DD21D3476
**Isn't it really an amazing coincidence that Earth's water fills the ocean beds so neatly, just barely lapping over the edges of the continental shelves?