Preheat the oven to 425 degrees F.
Place the chicken in a large bowl. Season all over with salt and pepper. Add the garlic, parmesan, breadcrumbs and herbs. Mix gently until just combined.
Roll the mixture into 1-inch meatballs.
Heat the olive oil in a large oven-safe skillet over medium heat. Add the meatballs in a single layer and brown for 3 to 4 minutes. Flip gently and brown the other side for another 3 to 4 minutes. Shake the pan, tossing the meatballs around until browned all over. Transfer the meatballs to a plate.
Add the butter to the same skillet 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. Add in the tomato paste and stir, cooking for 5 minutes. The color will darken - this is where the flavor is!
As a note, since this recipe does not call for pasta water, I make a faux pasta water! This is 1 cup of water and 1/4 teaspoon cornstarch and a pinch of salt. I bring it to a boil and then take it off the heat. It works great!
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 meatballs into the sauce. Toss to coat. Place the fresh mozzarella on top.
Place the pan in the oven and bake for 10 to 15 minutes. Remove and top with the shaved parmesan, fresh basil and crushed red pepper. Serve as desired! I love to serve with garlic toasts.