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Combine the basil, pine nuts, lemon juice, garlic, salt and pepper in a food processor. Pulse until finely chopped. With the motor running, slowly drizzle in the oil until well combined; set aside.
Pat each scallop dry with a paper towel. Cut each slice of prosciutto into 4 long strips. Wrap a strip of prosciutto around each scallop and secure with a toothpick.
Turn the grill on. Select the seafood cooking program and press OK. Once the purple indicator light has stopped flashing, open the grill, place the scallops and close the lid.
Cook until the indicator light has changed to orange. Top each scallop with a dollop of pesto and a few additional pine nuts.