Add the yeast and warm water to a small bowl and stir to combine. Set aside until foamy, about 5 to 10 minutes.
Whisk together the almond flour, psyllium husk powder, and baking powder in a medium bowl and set aside.
Add the part-skim mozzarella and cream cheese to a large microwave-safe bowl. Microwave for 60 seconds and then give it a stir, and continue microwaving in 20-second increments until the cheese is fully melted and combined when stirred.
Stir the yeast mixture into the melted cheese until combined, and then stir in the beaten egg until combined. Stir in the almond flour mixture until it forms a dough.
Oil your hands and knead the dough a couple times until it comes together as a ball. Divide the dough into 8 equal balls. Roll each dough ball out between 2 pieces of parchment paper to a circle about 5 inches in diameter.
Divide the mushroom filling between the 8 dough circles, leaving a 1-inch border. Place 1/2 tablespoon cottage cheese on top of the mushroom filling on each dough circle.
Fold each dough circle over so it looks like a half-moon, lightly pressing the outside together to seal it.
Place the pierogis onto a plastic-wrap lined plate and cover with plastic wrap. Refrigerate at least 12 hours, but up to 3 days.