Preheat oven to 350 degrees. Spread nuts on a rimmed baking sheet; toast in oven until golden and fragrant, tossing occasionally, 5 to 8 minutes. Let cool completely.
In a food processor, combine toasted nuts, basil, Parmesan, and garlic; season with salt and pepper. Process until finely chopped. With machine running, pour olive oil in a steady stream through the feed tube; process until smooth.
If not using immediately, the sauce should be refrigerated; cover with plastic wrap, pressing it directly on the sauce to preserve the bright color. This sauce will keep for up to three days.
Martha made this recipe for pesto as a delicious addition to a prosciutto and pesto panini. She used 1/3 cup pine nuts -- toasting them for 5 to 8 minutes -- and added a pinch of red-pepper flakes while processing.