Heat the oven to 375°F. Lightly grease 8 cups of a standard muffin tin.
Wrap the grated zucchini in a clean kitchen towel and squeeze out excess moisture. This helps the muffins hold together instead of turning watery.
Add the turkey, zucchini, carrot, egg, breadcrumbs, 2 tablespoons of the ketchup, garlic powder, salt, and black pepper to a mixing bowl. Mix gently until combined, taking care not to compact the meat.
Divide the mixture among the 8 prepared muffin cups. Spread the remaining 2 tablespoons of ketchup over the tops.
Bake until the centers reach 165°F and the tops are lightly browned, about 20 to 25 minutes. Let the muffins rest in the tin for 5 minutes before removing.
While the muffins bake, place the sweet potato chunks in a saucepan and cover them with water. Bring to a boil, then cook until the potatoes are very tender when pierced with a fork, about 15 to 20 minutes.
Drain the sweet potatoes and return them to the saucepan. Add the milk and butter, then mash until smooth. Serve the mash alongside the turkey muffins.