Make the pesto: In a food processor or blender, combine the basil, garlic, and pine nuts and pulse until coarsely chopped.
With the machine running, drizzle in the 1/2 cup extra-virgin olive oil until smooth, then add the 1/2 cup grated Parmesan-Reggiano and pulse to combine; season with salt and freshly ground black pepper to taste.
Cook the gnocchi according to the package instructions, then drain thoroughly 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 golden and crisp on the bottom, about 3–4 minutes.
Turn the gnocchi and continue to 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 to coat the gnocchi; add more pesto if desired.
Serve the gnocchi warm, topped with additional Parmesan cheese if you like.