In a large pot of salted boiling water, cook pasta until al dente. Drain, reserving 2 cups pasta water.
Meanwhile, in a large skillet over medium heat, melt butter. Add shallot and garlic and cook, stirring frequently, until softened, 4 to 5 minutes.
Add tomato paste and red pepper flakes and cook, stirring frequently, until shallots are coated and garlic begins to darken, 5 minutes.
Add vodka and stir to incorporate, scraping up any browned bits from the bottom of the skillet. Add heavy cream and ¼ cup pasta water, stirring to combine. Add ¼ cup Parmesan and stir until melted. Turn off heat and stir in pasta.
Fold in remaining ¼ cup Parmesan, adding more pasta water 1 tablespoon at a time to loosen the sauce if needed. Season with salt and serve topped with more Parmesan and torn basil.