Preheat the oven to 400 degrees F (200 degrees C).
Place sweet potatoes in a pot and add water to cover. Bring to a boil and cook for 15 minutes. Add cauliflower and more water if needed; cook until all vegetables are tender, about 15 minutes more. Drain. Mash sweet potatoes and cauliflower with 1/4 cup broth, butter, 1 teaspoon salt, and 1/2 teaspoon black pepper.
Heat olive oil in a large skillet over medium-high heat. Add carrots and onion; saute until softened, about 5 minutes. Add garlic, 1 teaspoon salt, and 1/2 teaspoon black pepper; cook 5 minutes more. Add beef; cook and stir until browned and crumbly, 5 to 10 minutes.
Sprinkle tapioca starch over the beef mixture in the skillet. Add 1 cup broth, peas, corn, and tomato paste. Bring to a boil. Reduce heat and simmer for 10 to 15 minutes. Spread beef mixture evenly in a baking dish. Top with mashed sweet potato mixture. Sprinkle Parmesan cheese on top.
Bake in the preheated oven until slightly browned on top, about 25 minutes. Let stand 15 minutes before serving.