Step 1

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.

Step 2

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.

Step 3

Do Ahead: Vodka sauce can be made 5 days ahead; cover and chill.