How to Make It
Preheat the oven to 425°. Rub the potatoes with the oil. Bake for 1 hour, until tender. Let cool slightly, then cut each potato crosswise into 4 thick slices.
Melt 2 tablespoons of the butter in a skillet. Add the potatoes and cook over moderate heat, turning, until golden and crisp, about 8 minutes. Season with salt; transfer to a paper towel-lined plate.
In a 7-inch omelet pan, melt 1/2 tablespoon of butter. In a bowl, beat 3 eggs, 1 teaspoon of Parmesan and a pinch each of salt and pepper and add to the pan; cook until just set, 2 to 3 minutes. Flip the omelet, set a burrata slice in the center and fold the sides over. Slide onto a plate. Repeat to make 3 more omelets and serve with the potatoes.