Bring a large pot of lightly salted water to a boil. Cook ziti in the boiling water, stirring occasionally until cooked through but firm to the bite, 8 minutes. Drain and transfer pasta to a bowl and toss with spinach and sour cream.
Heat a large skillet over medium heat; cook and stir sausage, onion, carrots, and garlic until onion is translucent and sausage is no longer pink, about 7 minutes. Add spaghetti sauce and broccoli to sausage mixture and simmer until sauce mixture is heated through, about 15 minutes.
Layer 1/2 of the ziti mixture into a 9×13-inch baking dish. Spread 1/2 of the fresh mozzarella cheese, 1/2 of the sauce mixture, the remaining ziti mixture, and fresh mozzarella cheese, respectively, into the baking dish. Sprinkle shredded mozzarella cheese over the dish. Cover dish with aluminum foil.
Bake in the preheated oven until cheeses have melted, about 20 minutes. Remove the aluminum foil and cook until top is lightly browned, about 10 minutes more.