Preheat the oven to 400°F (200°C). Lightly grease a deep 9-inch x 13-inch baking dish. You'll also need a sheet pan to place the baking dish on while cooking.
Place the sweet potatoes in a large pot and cover with cold, salted water. Bring to a boil, then reduce the heat and simmer for 15–20 minutes, until fork-tender. Drain thoroughly and return to the warm pot. Let stand for 2 minutes so excess moisture can evaporate.
Add the butter, milk, salt, and pepper. Mash until smooth, adding a little extra milk only if needed. Keep the mash thick enough to hold its shape.
Meanwhile, heat the olive oil in a large skillet over medium-high heat. Add the ground turkey and cook for 6–8 minutes, breaking it into small pieces, until lightly browned.
Add the onion, carrots, and celery. Cook for 7–9 minutes, stirring occasionally, until softened. Stir in the garlic, tomato paste, and poultry seasoning. Cook for 1 minute.
Sprinkle the flour over the filling and stir for 1 minute. Gradually stir in the broth, followed by the Worcestershire sauce, salt, and pepper. Bring to a simmer and cook for 5–7 minutes, until the gravy thickens. Stir in the frozen peas. The filling should be moist and spoonable, with no watery broth pooling around it.
Transfer the filling to the prepared baking dish. Spoon small dollops of sweet potato mash over the surface, then gently spread into an even layer. Drag a fork across the topping to create ridges.
Place the dish on a baking sheet and bake, uncovered, for 20–25 minutes, until bubbling around the edges and the centre reaches 165°F (74°C). Broil for 2–3 minutes if desired, watching closely, to lightly brown the ridges.
Let rest for 10 minutes before serving to help the filling settle.