Preheat oven to 350 degrees F (175 degrees C).

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 9x13-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.