Heat the olive oil in a large skillet over medium-high heat. When the oil is hot, add the chorizo, garlic and onion and cook until the chorizo is cooked through, 5 to 8 minutes. Drain any excess grease if needed.
Add the roasted bell peppers, corn and green chiles. Stir to combine. Add the cream and stock and bring to a boil. Allow the sauce to thicken slightly while you cook the gnocchi.
Drop the gnocchi into the boiling water and stir. Cook briefly, about 1 minute; the gnocchi will float to the top when ready. Using a slotted spoon, transfer the gnocchi to the sauce, then toss to combine.
Remove the skillet from the heat and add the cheese and butter. Continue to toss until the cheese has melted and everything has come together. Serve garnished with sliced green onions.