In a large skillet or Dutch oven, cook lamb (or beef), onion, and garlic over medium heat until no pink remains. Drain any fat.
Stir in flour, thyme, salt, and pepper and cook for 1 minute.
Stir in broth, tomato paste, and Worcestershire sauce and bring to a boil. Simmer uncovered 2 to 3 minutes or until thickened. Stir in mixed vegetables and add more broth if needed to reach a gravy consistency.
Transfer the ground beef mixture to a 2-quart casserole dish and spread it into an even layer. Spoon mashed potatoes over top.
Bake for 25-30 minutes or until bubbly. Broil for 2 minutes if desired.
Remove from the oven and cool for 5 to 10 minutes before serving.