Two 28-ounce cans whole tomatoes
2 tablespoons extra-virgin olive oil
2 tablespoons tomato paste
1 large basil sprig plus 2 tablespoons thinly sliced basil leaves
1/4 teaspoon crushed red pepper
4 garlic cloves, minced
2 pounds ground beef chuck
3 tablespoons dry bread crumbs
1 egg, lightly beaten
3 tablespoons freshly grated Parmigiano-Reggiano cheese, plus more for serving
Kosher salt and freshly ground pepper
Using your hands, crush the tomatoes into a slow cooker and add their juices. Add the olive oil, tomato paste, basil sprig, crushed red pepper and half of the garlic. Cover and cook on high for 3 hours, until the tomato sauce is very flavorful.
Meanwhile, in a large bowl, mix the ground chuck with the remaining garlic and the bread crumbs, egg, 3 tablespoons of Parmigiano-Reggiano, 1 tablespoon of kosher salt and 1/2 teaspoon of pepper. Roll the mixture into 18 meatballs, being careful not to pack them too firmly.
Add the meatballs to the sauce. Cover and cook on high for 1 hour longer, until no trace of pink remains in the center of the meatballs. Discard the basil sprig. Add 1 tablespoon of the sliced basil to the sauce and season with salt and pepper. Spoon the meatballs into bowls, sprinkle with the remaining 1 tablespoon of sliced basil and serve, passing grated cheese at the table.
MAKE AHEAD The cooked meatballs can be refrigerated in the sauce for up to 2 days before serving.