Bring a large pot of salted water to a boil. Add the pasta and cook as the label directs. Reserve 1/2 cup cooking water, then drain.
Meanwhile, mix the beef, 1/4 cup parmesan, the breadcrumbs, egg, 1/2 cup basil, the minced garlic and 1/2 teaspoon salt in a large bowl with your hands until just combined. Form into 3/4-inch meatballs (about 20). Grate the tomatoes into a medium bowl; discard the skins.
Heat a large nonstick skillet over medium heat. Add the olive oil, then the meatballs. Cook, turning, until browned on all sides, 2 to 3 minutes. Add the sliced garlic and cook until just golden, about 1 minute. Gradually add the vodka, then the grated tomatoes and cream. Bring to a simmer and cook until the sauce thickens and the meatballs are cooked through, 8 to 10 minutes. Season with 1/2 teaspoon salt and a few grinds of pepper.
Add the pasta and the remaining 1/4 cup basil to the skillet; toss, adding the reserved cooking water as needed to loosen. Remove from the heat and add the remaining 1/2 cup parmesan; toss.