- Heat a large skillet over medium-high and cook ground beef for 6–8 minutes until browned. Drain excess fat, leaving 2 tablespoons. 
- Add diced onions and cook 4 minutes; stir in garlic and cook 30 seconds. 
- Sprinkle flour, then mix in tomato paste, Worcestershire sauce, thyme, and broth; simmer until thickened. Fold in vegetables and season. 
- Boil peeled potatoes 15–18 minutes until tender; drain and mash with butter, warm milk, half cheese, and egg yolks (optional). 
- Preheat oven to 400°F. Spread beef mixture in greased casserole dish, top with mashed potatoes, smooth evenly, and sprinkle remaining cheese. 
- Bake 25–30 minutes until golden and bubbling. Broil 2–3 minutes for extra browning. Rest 10 minutes before serving.