In a food processor, combine gluten-free flour and salt, then add cold butter cut into small pieces. Pulse until it resembles coarse crumbs. Incorporate the egg yolk and ice-cold water, pulsing until a dough forms. Wrap and chill in the refrigerator for 2 hours.

In a large skillet over medium heat, melt butter and sauté diced onions and carrots until tender, about 5 minutes. Add sliced mushrooms and minced garlic, cooking until softened, about 3-4 minutes. Sprinkle rice flour and stir, then gradually add chicken stock and cream until thickened. Fold in chicken, peas, salt, and pepper and set aside to cool.

Divide chilled dough into two larger pieces and two smaller pieces. Roll out larger pieces into circles about ¼ inch thick and place in pie dishes.

Spoon cooled filling into pie crusts. Roll out smaller dough pieces and place over filling. Pinch edges to seal and use a fork for decorations.

Refrigerate assembled pies for 15 minutes, preheat oven to 425°F. Brush tops with egg wash and cut slits for steam. Bake for 25 minutes, covering after 15 minutes to prevent browning.

Remove from oven, let cool for 10 minutes before serving.