To prepare filling: Heat 1 teaspoon oil in a large skillet or Dutch oven over medium-high heat.
Add onions and carrots; cook, stirring, until golden brown and tender, about 7 minutes. Transfer to a bowl.
Heat the remaining 2 teaspoons oil in the pan over medium high heat.
Add mushrooms and cook, stirring often, until browned and their liquid has evaporated, 5 to 7 minutes.
Return the onions and carrots to the pan.
Add 2 cups broth and bring to a boil; reduce heat to a simmer.
Mix cornstarch with the remaining ½ cup broth; add to the pan and cook, stirring, until the sauce thickens.
Stir in turkey (or chicken), peas, sour cream, salt and pepper. Transfer the filling to a 2-quart baking dish.
To prepare biscuit topping and bake potpie: Preheat oven to 400°.Whisk whole-wheat flour, all-purpose flour, sugar, baking powder, baking soda, salt and thyme in a large bowl.
Using your fingertips or 2 knives, cut butter into the dry ingredients until crumbly.
Add buttermilk and oil; stir until just combined.
Drop the dough onto the filling in 5 or 6 even portions.
Set the baking dish on a baking sheet.
Bake the potpie until the topping is golden and the filling is bubbling, 30 to 35 minutes.
Let cool for 10 minutes; serve.