Melt butter in a large skillet over medium heat. Add mushrooms and scallions; cook, stirring occasionally, until softened, about 10 minutes. Transfer to a bowl.
Stir brown rice, sour cream, dill, salt, and pepper into the mushroom mixture in the bowl.
Unfold 1 puff pastry sheet onto an ungreased baking sheet. Spread mushroom mixture evenly on top, leaving a 1-inch border on all sides. Lay the second puff pastry sheet on top. Seal edges by pressing together with a fork.
Bake in the preheated oven until pastry is puffed up and golden brown on top, about 30 minutes. Cool slightly before serving, about 10 minutes.