Preheat the oven to 350°F and adjust the oven rack to medium position.

Heat the butter in a large skillet over medium heat. Add the bell pepper, onion, celery, and carrot and sauté the vegetables until tender, about 5 minutes. Make a well in the center of the vegetables and add the salt, pepper, and thyme. Stir the seasonings in the middle of the pan until fragrant, about 30 seconds, and then stir into the vegetables. Add the ground beef and stir and cook until beef is partially cooked, about 5-7 minutes.

Pour the beef filing into a 13 x 9-inch baking dish and spread evenly into a single layer.

Place the potatoes in a large pot, cover with water and bring to a boil over high heat. When the water begins to boil, reduce the heat to medium-high. Continue to boil until potatoes are cooked through, about 15-20 minutes. Drain the potatoes and then put them back into the pot. Add the milk, butter, sour cream and salt to the pot and using a potato masher, mash the potatoes until no large lumps remain. Season to taste with salt and pepper.

Spread the mashed potatoes over beef layer in baking dish. Sprinkle with cheddar cheese. Bake for 30 minutes or until bubbling and cheese is golden brown. Let cool for 10 minutes before serving.