Heat olive oil in a skillet over high heat. Add beef and cook until browned, about 2 minutes. Transfer the beef to a bowl and cover with plastic wrap to keep warm. Reserve the oil in the skillet.
Reduce heat to medium, then stir in the onion and garlic, and cook until transparent, about 5 minutes. Toss in the mushrooms, red peppers, and yellow peppers and cook until tender, another 5 minutes. Stir in the tomatoes and paprika; bring to a boil over medium-high heat. Reduce heat to low and simmer until sauce thickens, about 5 minutes. Combine the beef with the tomato mixture and stir.
Meanwhile, bring a large pot of lightly salted water to a boil over high heat. Add gnocchi and cook until they float, 2 to 3 minutes. Drain, then toss the gnocchi in the sauce and serve.