Of course they don't reveal their neat trick now because from a strategic point of view it would be bad, because other people would copy them, and probably outpace them.
Once it's only a physics problem, it becomes quite easy. Here are two potential solutions to the physics problem : Liquid mirrors, and plasma mirrors. The easiest being the liquid mirrors so let's explain this one. In fact it's so easy, it's child play.
Have your kids ever played with soap bubble : It's the same thing but in space. You make a big ring (roughly of the same area of the spot you are trying to make on the ground) (thin circular wire that you fold so that it will unfold itself like these tents) and you flow liquid metal (like mercury) on top to form a thin layer, hold by surface tension (and eventually electric field to have finer control of the shape or the surface).
The mirror will be a few atoms thick, a design that must be fabricated in space directly. It's just that we are not used to thinking with low force zero gravity environment. You don't use motors to move your mirror, you use fields spawned from precisely positioned points (and you have to compensate for solar winds).
Of course there are still technological problems due to the harsh conditions of space, but it's not something unheard of : https://engr.ncsu.edu/news/2024/03/06/reflecting-the-future-...