Add the pasta to a large pot of heavily salted boiling water. Cook according to the package instructions.
In a large Dutch oven over medium heat, melt the butter. Add the onion and shallot and cook, stirring occasionally, for 2 minutes. Add the garlic. Continue to cook, stirring, until the onion is translucent, an additional 2 to 3 minutes. Stir in the tomato paste and chile paste and cook until the tomato paste begins to caramelize and slightly stick to the pot, 2 to 3 minutes. Deglaze the pot with the vodka, releasing all the flavorful bits from the bottom. Continue to cook for 1 more minute. Add the heavy cream and stir to combine. Remove from the heat.
Reserve 1/2 cup of the pasta water for later use, then drain the pasta. Add the pasta directly to the pot of sauce and toss to combine. Add 1/4 cup of the pasta water and 1/4 cup of the cheese. Continue to mix, slowly adding more pasta water as needed a splash at a time, until the desired sauce consistency is achieved. (Discard any remaining pasta water.) Top with fresh basil and more cheese.