Freeze cubed butter for 10 minutes. Mix sour cream, vinegar, and water in a bowl and refrigerate.
Pulse gluten-free flour, sugar, and salt in a food processor. Process frozen butter into the flour mixture until evenly distributed.
Add the liquid mixture to the processor and pulse until clumps form. Add more water if needed.
Shape dough into a 1-inch thick disk, wrap in plastic, and refrigerate for 1 hour.
Preheat oven to 400°F (200°C). Roll out half the dough and line a pie dish.
In a pot, melt 4 tablespoons butter. Cook onion, carrots, and celery until softened.
Stir in gluten-free flour, chicken, broth, milk, and spices. Simmer until thickened. Add peas and cool slightly.
Pour filling into pie shell. Top with remaining dough, crimp edges, and brush with egg wash.
Bake for 35–40 minutes until golden. Cool slightly before serving.