Place the potatoes and carrots in a large pot with enough water to cover. Bring to a boil and cook until tender, about 15 minutes; drain. Separate the potatoes and carrots. Place the drained potatoes back into the pot. Add milk and 2 tablespoons butter. Mash the potatoes, adding salt and pepper to taste.
Melt the remaining 4 tablespoons butter in a large frying pan over medium heat. Stir in the onions and saute until softened, about 10 minutes.
Increase heat to medium-high and add the ground beef. Cook and stir until meat is no longer pink and begins to brown; drain excess fat. Stir in the tomato-basil sauce.
Place the carrots in the bottom of an ungreased pie dish. Add the beef mixture evenly over the top of the carrots. Spread the mashed potatoes evenly over the top of the beef mixture.
Bake uncovered until the internal temperature reaches 165 degrees F, about 20 minutes.
