A few days ago, I had a dream, the only part of which I can remember is a map.

The projection didn't make sense. It was rectangular, and latitude and longitude lines were all straight. However, features in polar regions weren't distorted.

I believe it was a sword and planet setting. The terrrain resembled Mars, with a mix of fluid-created landforms and craters. Some surface water, though not nearly as much as Earth or indeed most terraformed Mars concepts.

Most of the eastern hemisphere was one country. I know this was the "good" of at least "neutral" country, the default origin of the heroes for a story on this planet. The western hemisphere had many smaller countries. These would be the "exotic" foreign settings for the heroes to travel to.

I don't remember how I could tell climate from the map (as I recall, countries were color-coded, leaving no room for naturalistic colors), but the predominant temperature gradient was east-west, not north-south. The eastern hemisphere was overall much colder than the west. I recall that I liked this climate division: arctic-like climate wasn't something "other", it was the default/reference point.

Is it possible to build a world like this?

Tidal lock to the star is too extreme. I want both hemispheres to be inhabitable by humans. Also, there must be day and night everywhere.

I remember a concept from an astronomy book I have. A brown dwarf could plausibly have a moon?planet? as large as Earth, and it would probably be tidally locked. The near side would be warmer.

Not good enough; there's still a perpetual night side.

Try a star/brown dwarf binary with the world in question orbiting the brown dwarf. Now the far side gets light, with a "day" length determined by its orbit about the brown dwarf. This could be rather short. The Galilean moons have periods from 1.77 to 16.7 days, and a moon(?) close to the much more massive brown dwarf would orbit faster.

I suppose the question is whether I can construct a combination of star mass, binary separation, brown dwarf mass, and orbital distance suitable to create the climate I want. And then there's the question of exactly how hot and cold I want the "west pole" and "east pole" to be.