Mix the oat flour, millet flour, tapioca starch, salt, psyllium husk, and baking powder in a medium bowl.
Add the cold lard and cut it into the flour until mostly pea-sized pieces remain.
Stir the apple cider vinegar into the ice water. Add the egg white and about 3 tablespoons of the water mixture to the flour, adding more water as needed until the dough holds together.
Gently knead the dough into a ball, pat it into a rectangle on lightly floured baking paper, wrap tightly, and refrigerate for at least 30 minutes.
Mix the reserved egg yolk with a splash of water and refrigerate until needed.
Heat 1 tablespoon olive oil in a large pot over medium-high heat. Cook the chicken for 3 to 5 minutes, until no longer pink, then remove from the pot.
Add the remaining olive oil, celery, carrots, onion, and a splash of water. Cook for 5 to 8 minutes, stirring often, until slightly softened.
Return the chicken to the pot, sprinkle with chickpea flour, and stir to coat.
Add the chicken broth, cashew milk, thyme, and sage. Bring to a simmer and cook for 5 to 10 minutes, stirring occasionally, until thickened.
Remove from the heat, stir in the peas, and season with salt and pepper. Pour into a deep pie dish and cool for 30 minutes.
Preheat the oven to 420°F. Roll the chilled dough into a 12 x 8-inch rectangle, fold into thirds, and repeat the rolling and folding once more.
Shape the dough into a disk and roll into a circle a few inches wider than the pie dish. Place over the filling, trim to a 1-inch overhang, fold and crimp the edges, then brush with the egg yolk mixture and cut a few slits in the center.
Bake at 420°F for 30 to 40 minutes, until deeply golden and crisp. Cool for 10 minutes before slicing and serving.