Cut the pasta sheets into 3-by-3-inch squares. Brush one side of a pasta square with a small amount of egg, then place a piece of burrata in the center and top with a basil leaf. Season with salt and pepper, then top with one more pasta square to make a ravioli. Repeat to form a total of 16 ravioli for 3 portions. Allow the ravioli to set for 30 minutes in the fridge.
In the meantime, bring a pot of salted water to a boil. Bring the tomato sauce to a simmer in a separate saucepan.
Add the ravioli to the boiling water and cook until tender, about 3 minutes.
Add the olive oil to the tomato sauce and stir together. Add the ravioli to the pan with the sauce and cook for 1 more minute, adding the butter at the same time. Season with salt and pepper.
Divide the ravioli among 3 plates and garnish with extra basil and the Parmigiano-Reggiano.