Place the chicken breasts on a clean cutting board. Cover with foil or plastic wrap and pound until they are all the same thickness, about 1/2-inch thick. Dry them off with a paper towel and seasoned generously with salt and pepper on both sides.
Preheat the oven to 350 degrees F.
Heat a large oven safe skillet over medium high heat. Season with 2 swirls of olive oil. Once the oil is simmering, add the chicken breasts into the skillet and cook until lightly golden brown, about five minutes. Flip and cook again for five minutes. Remove from the skillet and adjust the heat to medium.
Add the butter and onion. Season with a generous pinch of salt. Cook for 5 to 7 min until softened.
Add the garlic and cook until fragrant, about 1 to 2 min.
Add the tomato paste. Toast the tomato paste for 5-7 minutes until it smells sweet like sun-dried tomatoes. Deglaze the pan with vodka. Scrape the golden brown bits off the bottom.
Add the heavy cream and stir until combined. Add the chicken stock. Taste and adjust salt levels if needed!
Place the chicken back into the skillet. Spread the sauce on top of the chicken if you can. Top each chicken breast with burrata and sprinkle fresh Parm on top.
Bake for 20-30 min until the chicken is fully cooked (165 degrees, internally) and the cheese is melty.
If your breadcrumbs aren't already toasted, toast them while the chicken is baking. Heat a skillet over high heat. Add the breadcrumbs and toast, shaking the pan every 30 seconds, until the breadcrumbs are a deep golden brown in color. Remove from the heat.
Bake for 15-20 min until the chicken is fully cooked and the cheese is melty.