Thanks! :)
In this game, all elements are made out of very simple shapes. Everything is either a line, circle, or a circle arc. I'm even using a ton of circle arcs to render the text, which is a really bad idea and horribly inefficient :D This solution worked for the simple shapes in this game, but it won't work in general, especially once you want to texture anything.
If you only want to show a section near the planet's surface, then you might want to check out @eevee's Lunar Depot 38. They used a shader to deform the world. The game is open source, the shader is here: https://github.com/eevee/lunar-depot-38/blob/master/ld38/scenes/world.lua#L41. But I'm not sure if their approach would work well if you want to show the entire planet...