Make the pesto: In a food processor or blender, combine the basil, minced garlic, and pine nuts and pulse until coarsely chopped.
With the machine running, slowly stream in 1/2 cup extra virgin olive oil and process until smooth.
Add 1/2 cup grated Parmesan-Reggiano and pulse until combined; season with salt and freshly ground black pepper to taste. Set the pesto aside.
Cook the gnocchi according to the package instructions in a large pot of boiling salted water; drain well in a colander.
Heat 2 tablespoons olive oil in a large skillet over medium-high heat until shimmering.
Add the drained gnocchi in a single layer and cook undisturbed until the bottoms are golden and crisp, about 3–4 minutes.
Turn the gnocchi and cook, turning occasionally, until all sides are golden and crisp, about 3 more minutes.
Remove the skillet from the heat and stir in 1/4 to 1/3 cup of the prepared pesto until the gnocchi are evenly coated. Reserve any extra pesto for another use.
Serve the crispy gnocchi warm, topping with additional Parmesan cheese if desired.