Combine the basil, garlic, and pine nuts in a food processor or blender and pulse until coarsely chopped. With the machine running, slowly add the olive oil and process until smooth. Add the cheese and pulse until combined. Season with salt and pepper, to taste. Set aside.

Cook the gnocchi according to package instructions. Drain completely. In a large skillet, heat olive oil over medium-high heat. Add the gnocchi in a single layer and cook, undisturbed, until golden and crisp, about 3-4 minutes. Turn the gnocchi and cook, turning occasionally, until golden and crisp, about 3 more minutes. Remove from the heat and stir in 1/4 to 1/3 cup of the pesto. You might have leftover pesto. Serve warm and garnish with Parmesan cheese, if desired.