Heat oil in a large, heavy saucepan over medium-high heat. Add mushrooms and garlic and saute until the mushrooms are just tender, 3 to 5 minutes. Sprinkle flour over the mushrooms and cook, stirring, for 1 minute. Pour in wine, stirring with a whisk. Gradually whisk in broth and evaporated milk and bring to a boil, stirring. Cook, stirring, until thickened and smooth, about 1 minute. Remove from the heat and stir in chicken (or turkey), lemon juice, parsley and thyme. Season with salt and pepper.
Blanch carrots in boiling water until tender, about 3 minutes. Stir the carrots, onions and peas into the filling. Spoon the filling into a deep, 3-quart casserole dish and set aside while preparing biscuit topping.
Preheat oven to 400°F.
Make Light & Fluffy Biscuit dough. Pat the biscuit dough into a ½-inch-thick disk. Cut out rounds with a 3-inch cutter. Gather the dough scraps together and repeat. Cover the filling with biscuit-dough rounds and brush with fat-free milk.
Bake for 20 minutes. Reduce the oven temperature to 350°F and bake until the biscuits are nicely browned and the filling is bubbling, an additional 10 to 20 minutes. Serve immediately.
Make Ahead Tip: Cover and refrigerate the filling (Steps 1-2) overnight.