Sauté aromatics: In a pan or pot over medium heat, add olive oil and sauté the chopped onion for 5 minutes until translucent. Add the minced garlic and cook for 3 more minutes, stirring frequently.
1 tablespoon olive oil, ½ yellow onion, 3 cloves garlic
Make the vodka sauce: Add the tomato paste and spread it slightly with a spatula. Cook for 1 minute, stirring occasionally, until lightly caramelized. Pour in the vodka to deglaze the pan, mixing well. Add the milk and whisk until fully combined. Stir in the kosher salt and let simmer for 1 minute.
¼ cup tomato paste, ⅓ cup vodka, 1 cup milk, 1 teaspoon kosher salt
Add the beans: Pour in the drained white beans and chickpeas. Add butter if using. Raise the heat to medium-high and cook for 5–7 minutes, stirring occasionally, until the sauce has reduced slightly. The sauce should bubble lightly. If you notice heavy bubbling, reduce the heat.
1 (15oz can) white beans, 1 (15oz can) can chickpeas, 1 tablespoon butter
Finish and garnish: Lower the heat and stir in the Parmesan. Remove from heat. Sprinkle the mozzarella and additional Parmesan. Blend pesto in a blender first for a smooth swirl if desired. Add pesto to a squeeze bottle, then swirl it over the cheesy beans. Alternatively, dollop small bits of the pesto all over the cheesy beans. Garnish with fresh herbs and serve with bread for dipping.
¼ cup grated Parmesan, ⅓ cup pesto