Preheat the oven to 350 degrees. Grease a 2-quart baking dish with nonstick cooking spray.
In a large pot of salted water, boil the potatoes until very tender.
Meanwhile, heat the oil in a large nonstick skillet. Add the beef, onion, carrots, and mushrooms and cook until the beef is browned and the onion is translucent. Add the flour and stir until the drippings have completely absorbed the flour. Stir in the broth and spices. Simmer until thickened. Stir in the green beans or peas. Spoon this mixture into the prepared baking dish.
Once the potatoes are cooked, drain them and return them to the pot. Mash the potatoes until very fine. Add the butter and continue mashing, until the butter is melted and incorporated into the potatoes. Stir in the milk and cheddar cheese. Season to taste with salt and pepper.
Spoon the mashed potatoes over the meat mixture, taking care to seal the mashed potato topping all the way to the edges of the pan.
At this point you can chill the pie and refrigerate to bake later or chill, cover and freeze up to 2 months. Before baking, thaw the pie in the refrigerator .
Bake the pie for 30 minutes or until the filling is bubbly and the top is golden brown. If the pie is very cold, you may need to increase the baking time up to an hour.