Start the topping: Place potatoes in a large pot and cover with salted water; bring to a boil. Reduce heat to medium-low and simmer until tender, 15 to 20 minutes.

While the potatoes are cooking, preheat the oven to 375 degrees F (190 degrees C).

At the same time, make the filling: Heat olive oil in a skillet over medium heat; stir in onion. Sauté onion until soft and translucent, about 5 minutes. Stir in ground turkey, mushrooms, carrot, parsley, garlic, bouillon powder, and thyme. Cook and stir until meat is broken up and cooked through, 5 to 7 minutes. Season with salt and pepper. Stir in flour and cook for 1 minute more. Remove from the heat and set aside.

Finish the topping: Drain potatoes and transfer to a bowl. Add milk and butter and mash to desired consistency. Season with salt and pepper.

Transfer filling to a deep-dish pie pan or casserole dish. Spread topping over filling and swirl with a fork.

Bake in the preheated oven until the topping is lightly browned, about 30 minutes.