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

Bring a large pot of lightly salted water to a boil. Add ziti and cook, stirring occasionally, until tender yet firm to the bite, about 8 minutes. Drain and set aside.

Meanwhile, heat a large skillet over medium heat. Cook and stir ground beef, onion, and 1/2 of the garlic in the hot skillet until beef is browned and onion is wilted, about 5 to 7 minutes. Add spaghetti sauce and simmer for 15 minutes.

While meat sauce is simmering, heat olive oil in a medium skillet over medium heat. Add remaining garlic and cook until fragrant, about 1 minute. Add spinach and cook, tossing occasionally, until wilted, about 2 minutes. Remove from heat.

Mix together ricotta cheese, 1/4 cup Parmesan cheese, and wilted spinach in a large bowl until well combined. Season with salt.

Place 1/2 of the ziti in the bottom of a 9x13-inch baking dish and layer with 1/2 of the meat sauce. Spread provolone cheese on top to form a flat surface. Spread spinach mixture over the top, then cover with remaining ziti and meat sauce. Top with 2 tablespoons Parmesan and sprinkle mozzarella over top.