Preheat the oven to 425°F. Brush a small baking dish with some of the oil. Set aside.
For the butternut squash: Line 2 rimmed baking sheets with silicone mats or parchment paper. Place the squash on the prepared baking sheets in a single layer. Season with salt and pepper. Roast, rotating the baking sheets once, until the squash is just tender, about 10 minutes. (Leave the oven on after vegetables are cooked.) Remove the baking sheets to the stovetop to cool slightly. You can go ahead and combine the two pans once the squash is cooled just a bit.
For the béchamel: Heat 1 tablespoon of the oil in a small saucepan over medium heat. Add the flour and cook until foaming, about 1 minute. Add the milk and bring to a boil. Reduce the heat to medium and simmer until thickened, about 3 minutes. Add 1/4 cup of the Parmesan cheese. Season with salt and pepper. Set aside.
For the tomato-turkey sauce: Heat the remaining tablespoon of oil over medium heat in a large skillet. Add the turkey and season with salt and pepper. Cook stirring occasionally until the turkey starts to turn white. Add the tomato sauce, garlic, red pepper flakes, and nutmeg. Stir until combined and the turkey is cooked, about 10 minutes.
Place 2 cups of tomato-turkey sauce in the bottom of the prepared baking dish. Layer with half of the roast butternut. Top with the béchamel mixture, spreading with a spoon to cover the squash completely. Top with 1/2 cup of the tomato-turkey sauce and spread to cover. Layer with the remaining squash and top with the remaining cup or so of tomato-turkey sauce.
Top with grated mozzarella and sprinkle over the remaining tablespoon of Parmesan. Transfer to the oven and cook until bubbly and golden brown, about 30 minutes. Remove to a rack to cool slightly, then serve.