In a large skillet, heat oil over medium-high heat. Add onion and ground turkey. Cook until crumbly, about 5 minutes. Add the oregano and garlic, cook an additional minute. Pour meat into a strainer and set aside for a few minutes. Add skillet back to the heat. If the pan is a little dry, add a little more olive oil and return to medium-high heat.
Add gnocchi and sauté about 5-6 minutes, or until golden brown. Return the ground turkey to the skillet and mix in the beans, tomatoes and spinach. Cook for 2 more minutes. Add any salt and pepper at this time.
Turn off heat, sprinkle with basil and both cheeses. Cover for a few minutes. This will allow the cheeses to melt.