Heat a grill pan to high.
Toast the nuts lightly in small skillet over low heat, then add them to a food processor.
Add the arugula, basil, lemon juice, garlic, and salt and pepper, to taste, and pulse to combine.
With the processor on, stream in the extra-virgin olive oil to form thick pesto.
Lightly pound out the chicken, then rub all sides with the pesto. Spray the grill pan with cooking spray or rub with a little oil and grill the chicken 2 to 4 minutes on each side.
Cook's Note: Serve sliced on top of Caesar Spaghetti or, pile it onto ciabatta bread with sliced tomatoes and mozzarella for an outta-sight sammy.