In a large skillet over medium heat, melt 2 Tbsp. butter. Add onion; season with ½ tsp. salt and cook, stirring occasionally, until onion is sweet to the taste, 5 to 7 minutes. Stir in garlic and remaining 2 Tbsp. butter and cook, stirring, until butter is melted and garlic is fragrant, about 30 seconds. Add tomato paste and cook, stirring, until darkened and starting to stick to bottom of skillet, 3 to 4 minutes.
Add vodka and bring to a simmer. Cook, scraping browned bits from bottom of skillet, until liquid is reduced by half, 1 to 2 minutes.
Add cream and ¾ c. water, and stir until incorporated. Fold in beans; season with ½ tsp. salt. Return to a simmer and continue to cook, stirring occasionally, until sauce is slightly reduced and coats beans, 3 to 4 minutes more.
Remove from heat and fold in cheese until melted; season with salt, if needed. Top with basil and more cheese.
