Sauté the vegetables: Heat olive oil in a skillet over medium heat. Add onion, carrots, and garlic. Stir until softened, about 5 minutes.
Cook the turkey: Add ground turkey to the skillet. Break it up and cook until browned, about 8 minutes. Drain any excess fat.
Simmer the filling: Stir in tomato paste, Worcestershire sauce, thyme, and black pepper. Pour in turkey broth. Simmer for 15 minutes until thickened.
Boil the potatoes: While the filling simmers, boil potatoes in salted water for 15 minutes until fork-tender. Drain and return to the pot.
Mash the potatoes: Add milk and butter to the potatoes. Mash until smooth and creamy. Season with salt to taste.
Assemble the pie: Pour the turkey mixture into a baking dish. Spread mashed potatoes on top. Sprinkle cheese if desired.
Bake: Place in a preheated oven at 375°F (190°C) for 25 minutes until golden and bubbly.
Serve: Garnish with chopped parsley and serve hot.