In a large skillet or pot, heat oil over medium heat. Add shallot and cook, stirring until softened, about 3 minutes. Stir in garlic and hot pepper flakes and cook 1 minute.
Add tomato paste and cook, stirring until dark red and beginning to caramelize, 1 minute. Add vodka and cook until most of liquid is evaporated. Stir in cream, salt and bring to a low boil; remove from heat.
Meanwhile, cook pasta in a large pot of boiling salted water, following the package directions. Drain, reserving 1 cup of the cooking liquid. Add pasta and butter to skillet and cook over medium-low heat.
Stir in 1/2 cup of the reserved cooking liquid, stirring constantly, adding more of the liquid as needed until butter has melted and sauce is glossy and thickened, about 1 minute.
Sprinkle in 1/4 cup of the Parmesan, tossing to coat. Divide among bowls and garnish with the remaining Parmesan.