Preheat the oven to 400°F. Arrange a 9x9-inch (or similar) oven-safe dish and set aside.
Place the peeled, cut potatoes in a pot and cover with cold water. Add a generous pinch of salt, bring to a boil, then simmer until fork-tender, about 12–15 minutes.
While the potatoes cook, heat 1 tablespoon olive oil in a large skillet over medium-high heat. Add the chopped onion and diced carrots and sauté until softened, about 6–8 minutes.
Push the vegetables to the side, add the ground turkey to the pan, and break it up with a spatula. Cook until no longer pink, about 6–7 minutes. Stir in minced garlic and cook 30 seconds more.
Sprinkle the flour over the turkey-vegetable mixture and stir for 1 minute to cook the raw taste out. Add tomato paste, Worcestershire sauce, smoked paprika, and thyme; stir to coat.
Pour in the chicken broth, scraping up any browned bits from the pan. Bring to a simmer and reduce the heat; let the filling bubble gently until it thickens slightly, about 6–8 minutes. Fold in the frozen peas, season with salt and pepper, then remove from heat.
Drain the cooked potatoes and return them to the pot. Add butter, warmed milk, nutmeg (if using), salt, and mash until smooth and creamy. Stir in the Parmesan if using. Taste and adjust seasoning.
Spoon the turkey filling into the prepared baking dish and spread into an even layer. Dollop the mashed potatoes on top and use a spatula or fork to smooth and create shallow ridges (these help with browning). Brush the surface with the olive oil or melted butter.
Bake in the preheated oven for 20–25 minutes, until the edges are bubbling. For a crisp, golden top, switch the oven to broil for 2–3 minutes—watch carefully so it doesn’t burn.
Remove from the oven and let the pie rest for 8–10 minutes before serving to allow the filling to set. Serve warm.