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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 paste has coated shallots and garlic and is beginning to darken, about 5 minutes.
Add vodka and stir to incorporate, scraping up any browned bits from bottom of pot. Remove from heat.
Meanwhile, in a large pot of boiling salted water, cook pasta, stirring occasionally, until al dente according to package directions. Drain, reserving 2 cups pasta water.
Return sauce to medium heat and add cream and ¼ cup pasta water, stirring to combine. Add half of Parmesan and stir until melted. Remove from heat and stir in cooked pasta. Fold in remaining Parmesan, adding more pasta water (about 1 Tbsp. at a time) if the sauce is looking dry; season with salt, if needed.
Divide pasta among bowls. Top with Parmesan and basil.