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 whole basil leaves and halved onion and simmer for 1 hour with the lid tilted, stirring occasionally.
After the simmering time, remove the onion. Carefully transfer the sauce to a blender, or use an immersion blender to blend the sauce.
Stir in the Parmesan cheese until it melts. Taste and adjust seasonings as desired. Serve immediately or refrigerate until serving. Makes enough to add to 1 pound pasta.