Preheat the broiler. Heat the olive oil in a medium ovenproof nonstick skillet over medium-high heat. Add the onion and carrot and cook, stirring occasionally, until softened, about 5 minutes. Add the garlic and cook until softened, 1 minute. Add the beef, 1/2 teaspoon salt and a few grinds of pepper and cook, stirring occasionally, until no longer pink, about 4 minutes. Add the tomatoes with their juices, oregano, 1/4 teaspoon salt and a few grinds of pepper. Bring to a boil, then reduce the heat to a simmer and cook until the sauce thickens, 15 to 20 minutes.
Meanwhile, bring a large pot of salted water to a boil. Add the gnocchi and cook as the label directs. Reserve 1/4 cup cooking water, then drain. Toss the gnocchi with the butter and reserved cooking water in a large bowl until coated; stir in the parsley and season with salt and pepper.
Spread the gnocchi in an even layer over the beef mixture in the skillet and sprinkle with the mozzarella. Broil until the gnocchi are browned in spots and the cheese is melted, about 3 minutes. Sprinkle with more parsley.