Bring a large pot of salted water to a boil. Cook pasta until just shy of al dente.
Scoop out about 1/2 cup pasta water, then drain the pasta.
Warm olive oil and melt butter in a wide skillet over medium heat.
Soften the shallot, stirring often, until translucent, 3 to 4 minutes.
Stir in garlic and cook until fragrant, about 30 seconds.
Add tomato paste and cook, stirring, until darker and caramelized, 3 to 5 minutes.
Pour in vodka. Simmer until the alcohol cooks off and it reduces by about half, 2 to 3 minutes.
Lower heat and stir in heavy cream until the sauce is smooth and rosy.
Add drained pasta to the skillet with a splash of reserved water. Toss until coated and glossy.
Take off heat. Stir in Parmesan until melted, loosening with more pasta water if needed.
Season with salt and black pepper. Fold in basil and serve with extra cheese.