Preheat your oven to 375°F (190°C) and then in a large oven-safe skillet or Dutch oven, brown the ground beef over medium-high heat, breaking it apart with a spoon.

Drain any excess grease, then add the chopped onion, diced carrots, diced celery, and minced garlic to the skillet, cooking until the vegetables soften (about 5-7 minutes).

Stir in the tomato paste and cook for 1 minute, then add the beef broth, Worcestershire sauce, dried thyme, salt, and black pepper, bringing the mixture to a simmer and cooking for 5-7 minutes until slightly thickened.

While the filling simmers, boil the peeled and quartered potatoes in a large pot of salted water until they are fork-tender (about 15-20 minutes).

Drain the potatoes, then return them to the pot and mash them with milk and butter until smooth, seasoning with salt and pepper to taste.

Spread the meat filling evenly in the oven-safe skillet or transfer it to a 9x13 inch baking dish.

Carefully spoon the mashed potato topping over the meat filling, spreading it to the edges.

Bake for 20-25 minutes, or until the topping is golden brown and the filling is bubbling.