In the bowl of a food processor combine the packed basil, 1/2 cup of the cheese, garlic, salt, and pepper, and pulse several times to chop the basil. While the processor is running, add 3/4 cup of the oil in a thin, steady stream. Add 1/4 cup of the pine nuts and the walnuts and pulse several times until the nuts are incorporated and chopped but pieces are still visible. Taste and adjust seasoning if necessary. Set aside. Makes about 1 cup.
Season the scallops on both sides with salt and freshly ground black pepper, to taste. Wrap each scallop around the smooth outer edges with a piece of prosciutto so that the entire outer circumference of the scallop is covered with prosciutto.
Bring a large pot of salted water to a boil and add the linguine. Cook according to package directions until al dente. Drain, toss with the remaining 1/4 cup of olive oil, and set aside to keep warm.
Melt half of the butter in a medium skillet over medium-high heat and, when the butter just begins to brown, add half of the scallops and cook until golden brown on both sides, about 1 to 2 minutes per side. Transfer to a plate and keep warm while you cook the remaining scallops. (You may need to wipe the skillet between batches if the butter gets too brown.)
Season the linguine with salt and pepper to taste and divide among 4 or 6 entree plates. Divide the scallops evenly among the plates, placing them directly on top of the linguine. Drizzle each plate of scallops with 2 to 3 tablespoons of the pesto. Garnish with remaining toasted pine nuts and fresh basil leaves. Pass the remaining grated cheese at the table.