In order to build the world of Westeros, we just need to scale it up. Instead of having a small asteroid-sized object orbiting a binary dwarf planet, we could have an Earth-sized world orbiting a twin gas giant. If you had, say, a Saturn-sized planet with a super-Earth orbiting it close by, or a massive gas giant orbited by the rocky core of Jupiter, any moons of those binary worlds — even Earth-sized moons — would have that same chaotic, tumbling behavior. Night and day will still be a reality on a world like this, as the Earth-sized moon would still rotate rapidly with respect to the Sun, but the rotational axis would be wildly unpredictable. This would cause large variations in both the onset of seasons and in the length of night/day, potentially even leading to months or years of darkness on end for part of the world.