Pull off a manageable portion of the dough, and roll out on a lightly floured surface until very thin—it should be thin enough to be translucent. The dough will be stubborn and try to pull back at first, but eventually it will allow itself to be rolled out thin.
Cut the dough into 4” or 10cm circles. You’ll need two for each stuffed sopaipilla. Arrange them in a single layer on a lightly floured surface.
Arrange 2 tablespoons of shredded mozzarella onto one of the dough rounds, topped with 1 tablespoon of chèvre. Leave a 1/2” to 3/4” (1.25 to 2 cm) rim around the outside without cheese on it.
Get your fingertips damp and moisten the clean rim of the cheese-topped dough round. Lay another dough round on top of it and seal the edges, gently squeezing out as much air as possible as you go.
Repeat for each sopaipilla. (I recommend making them all at once and setting them aside on a lightly floured surface while you finish, so you can fry them more quickly.)