Warm 4 tablespoons of olive oil in a heavy-bottomed pot over medium heat. Add 1 pound of ground beef and cook, breaking it apart, until well browned all over.
Stir in 1 cup of chopped onion and 3 chopped garlic cloves. Cook for about 2 minutes until the onion softens, then work 1 tablespoon of tomato paste through the meat mixture until fully combined.
Pour in 3 cups of diced ripe tomatoes and 2 cups of water. Stir well, bring to a boil, then reduce heat and simmer uncovered for 2 hours, stirring occasionally, until the sauce is thick and deeply flavored. Season with salt and pepper throughout.
Bring a large pot of salted water to a boil and cook 1 pound of spaghetti until just al dente. Drain well and set aside.
Spoon the bolognese over the drained spaghetti a little at a time, tossing until every strand is coated but not drenched. Fold in all 3 cups of grated hard cheese until fully incorporated.
Transfer the pasta mixture into a baking dish and spread evenly. Scatter ½ pound of shredded mozzarella across the entire surface. Bake in a preheated 350°F oven for 10 minutes or until the mozzarella is fully melted and lightly golden. Serve immediately.