As a note, since this recipe does not call for cooked pasta, I make a faux pasta water! This is 1 cup of water and ¼ teaspoon cornstarch and a pinch of salt. I bring it to a boil and then take it off the heat. It works great!
Heat the butter in an oven-safe skillet or dutch oven over medium-low heat. Once melted, stir in the shallots and garlic with a big pinch of salt and pepper. Cook, stirring often, until the onions soften, about 5 minutes. Add in the tomato paste and stir, cooking for 5 minutes. The color will darken - this is where the flavor is!
Stream in the vodka and stir, scraping the brown bits and flavor from the bottom. Stir in ¾ cup of the reserved pasta water. Stir in the heavy cream and parmesan, stirring until the cheese melts. Taste and season the sauce with more salt and pepper if needed. You can also stream in a bit more pasta water here if you want it slightly thinner!
Add the beans to the sauce and stir gently to combine. Top with the cheese.
Turn on the broiler. Place the pan directly under it and cook for 2 to 3 minutes, or until the cheese is golden and bubbly.
Remove the pan from the oven and top with extra parmesan cheese and the fresh basil. Serve!