- Preheat oven to 400ºF and lightly grease a 9-inch pie pan.
- Melt butter over medium heat in a large saucepan. Add onion, carrots, and celery. Sauté for 5 minutes. Stir in thyme, salt, pepper, potatoes, and mushrooms. Continue to cook for 5 minutes longer.
- Sprinkle flour over vegetables and stir well to coat. Gradually stir in broth, stirring constantly. Stir in Takumi sauce and almond milk. Bring to a simmer and cook until liquid has thickened, about 5 minutes. Stir in peas.
- Scoop mixture into prepared pan. Top with puff pastry and trim around the shape of the pan leaving ½ inch overlapping. Use a fork to press pastry edges to pan. Cut 3-4 slits in the center of the pastry for venting. Brush the pastry lid lightly with egg wash. Bake for 25-30 minutes, until crust is golden brown.
- Cool pot pie for 5 minutes before slicing into triangles. Enjoy hot.