Bring a large pot of salted water to a boil. Cook ravioli according to the package instructions. Before draining, reserve ½ to 1 cup of the starchy pasta water. Set ravioli aside.
While the ravioli cooks, zest one lemon and mince or grate the garlic cloves. Cut the lemon in half and set aside one half for the sauce later.
In a large pan over low to medium heat, melt the butter with a small drizzle of olive oil. Add the minced garlic and stir for about 1 minute until fragrant—do not let it brown. Add the lemon zest and stir for another minute, then squeeze in the lemon juice.
Add ½ to 1 cup of the reserved pasta water to the pan. Increase the heat to medium and let the sauce simmer until slightly reduced and glossy, about 2–3 minutes. Turn off the heat and stir in the grated parmesan until melted and combined. Taste and season with salt and pepper.
Plate the ravioli, spoon the lemon parmesan sauce over top, and gently nestle torn pieces of burrata around the pasta (make sure it's at room temperature so it melts slightly). Garnish with fresh basil, a sprinkle of flaky salt, black pepper, and extra lemon zest/grated parm if desired.