1/2 cup grated Parmesan cheese, plus more for garnish
Mince the garlic.
Melt the butter in large saucepan over medium heat. Add the garlic and cook until lightly browned, about 3 minutes. Add the tomatoes, vodka, oregano, red pepper flakes, and kosher salt. Stir to combine. Add the halved onion and simmer for 1 hour with the lid tilted, stirring occasionally.
After the simmering time, remove the onion. Transfer the sauce to a blender, or use an immersion blender to blend the sauce. Stir in the Parmesan cheese until it melts.
Meanwhile, boil heavily salted water in a large pot. Cook the pasta to al dente (see the section above). When the pasta is done, drain it and return it to the pot. Combine it with the vodka sauce. Serve sprinkled with grated Parmesan cheese and torn fresh basil leaves.