Bring 4 quarts water to boil in large pot. Meanwhile, heat oil in 12-inch skillet over medium heat until shimmering. Add the sliced garlic and pepper flakes and cook until fragrant, about 30-60 seconds. Add tomato sauce and cook until slightly thickened, about 3 minutes. Add vodka. Bring sauce to boil, reduce heat to medium, and simmer until thickened, 12 to 15 minutes. Stir in oregano and cream. Season with salt and pepper to taste; cover and keep warm.

Add pasta and 1 tablespoon salt to boiling water and cook, stirring often, until al dente. Reserve ¼ cup cooking water, then drain pasta and return it to pot. Toss sauce and basil with pasta, adding reserved pasta water as needed. Top with grated Parmesan cheese and serve.