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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 sea scallop dry with a paper towel. Cut each slice of prosciutto into 4 long strips. Wrap a strip of prosciutto around each sea scallop and secure with a toothpick.
Turn the grill on. Select the program Fish and press OK . Once the purple indicator light has stopped flashing, place the sea scallops on the grill and close the lid. Cook until the indicator light has changed to orange.
Top each sea scallop with a dollop of pesto and a few additional pine nuts.