It can be done, but there would be huge storms as a result.
The only natural phenomena that I can think about that would be able to produce this sort of result would be tidal locking. Basically what happens is that a planet is so close to its star that the gravity makes it spin at the same rate it orbits the star, making one face of the planet permanently fixed to the star. From the planet's perspective, it would be like a sun that never moves. Now, this has its complications, mainly the following:
1.- Having a tidal lock on a star like our sun would melt half of the planet so in order to be habitable the star would have to be a red dwarf: in other words, a very small and cold star.
2.- Even with a small star the temperature would be really high on the bright side. Just to put in perspective, the sun for such a planet would occupy probably at least half of the sky; on the equator, temperatures would be enough to melt rock. Probably the habitable part would be on the twilight between the dark side and the bright side.
3.- But talking of the dark side, it would be incredibly cold, freezing solid oxygen kind of cold due to its permanent shadow if it weren't due to the next point.
4.- You have atmosphere, and storms within atmospheres are driven mainly due to heat differences. When air is hot, it expands; when it's cold, it contracts; and when a cold mass of air crashes with a hotter mass of air, they start to combine, expanding and contracting respectively creating something called eddie currents, which are common to what we call a tornado. On Earth, this happens when the sun heats the sea and its hot air crashes with the cold air coming from the poles, and the greater the difference the greater the force of the storm. Now on your world we are not talking just about a small gradient like here on Earth. We are talking about air as hot as melting rock crashing with air cold enough to freeze oxygen; there would be global storms stupendously huge engulfing the planet all the time, making the dark side somewhat warmer and the bright side somewhat colder than they would be otherwise, but the air speeds would be just amazing, probably in the hundreds of mph. Actually there are some scientific studies on this.