Get a large pot of generously salted water on the stove over high heat and bring it to a rolling boil. Meanwhile, melt butter with olive oil in a large saucepan (at least 12-inch) over medium heat, then add the minced garlic. Cook for 2-3 minutes, stirring frequently, until the garlic becomes soft and fragrant but hasn't turned golden or brown.

Add the tomato paste and Calabrian chili paste to the pan, stirring continuously to incorporate everything. Keep cooking and stirring for 3-4 minutes until the tomato paste darkens to a deeper brick-red color. This caramelization develops much richer flavor.

Pour in the vodka and stir vigorously for 2-3 minutes, scraping up any flavorful bits stuck to the bottom of the pan. The vodka will deglaze the pan and help emulsify the sauce as the alcohol cooks off. You'll notice the sauce coming together as it reduces slightly.

Reduce heat to medium-low and pour in the heavy cream, stirring until fully incorporated. Let this simmer gently for 3 minutes, then add a hefty pinch of sugar, a pinch of salt, and 1/2 cup of the grated Parmesan. Stir until the cheese melts completely into a smooth sauce, then reduce to low heat.

Drop the ravioli into your boiling water and cook for just 2 minutes or until they float to the surface and puff up slightly. Don't overcook them since they'll finish cooking in the sauce. Before draining, scoop out 1 cup of the starchy pasta water and set it aside.

Drain the ravioli and add them directly to the vodka sauce. Pour in 1/2 cup of the reserved pasta water and stir gently to avoid breaking the ravioli, working over medium-low heat until the sauce becomes silky and coats each piece beautifully. If it looks too thick or dry, add more pasta water a tablespoon at a time.

Transfer to serving plates or a large bowl, then tear the room-temperature burrata into pieces and arrange on top. Finish with a sprinkle of salt, freshly cracked black pepper, and the remaining Parmesan cheese. Serve immediately while the burrata is melting.