Heat up pasta water to a boil before starting on the sauce. Don't forget to salt the water! Cook the pasta to al dente and reserve 1 cup of pasta water to help the sauce later.
Heat oil in a large skillet over medium. Add shallot and garlic and cook, stirring occasionally, until softened, about 5 minutes.
Add tomato paste and cook, stirring occasionally, until paste is brick red and starts to caramelize, about 5 minutes.
Add vodka and cook, stirring constantly, until liquid is mostly evaporated, about 2 minutes.
Add cream and red pepper flakes and stir until well blended. Season with salt and pepper; remove from heat.
Meanwhile, cook pasta in a large pot of boiling salted water, stirring occasionally, until al dente. Drain, reserving 1 cup pasta cooking liquid. Add pasta to skillet with sauce along with butter and ½ cup pasta cooking liquid. Cook over medium-low heat, stirring constantly and adding more pasta cooking liquid if needed, until butter has melted and a thick, glossy sauce has formed, about 2 minutes. Season with salt and pepper and add 1 oz. Parmesan, tossing to coat. Divide pasta among bowls, then top with basil and more Parmesan.
Add cooked pasta to skillet with sauce along with butter and ½ cup pasta cooking liquid. Cook over medium-low heat, stirring constantly and adding more pasta cooking liquid if needed, until butter has melted and a thick, glossy sauce has formed, about 2 minutes.
Season with salt and pepper and add 1 oz. Parmesan and basil, tossing to coat. Divide pasta among bowls, then top with basil and more Parmesan.