1 16-ounce bag frozen shelled edamame (soybeans)
1/2 cup coarsely chopped fresh basil
1/3 cup extra-virgin olive oil plus additional for brushing
3 garlic cloves, peeled
6 tablespoons freshly grated Parmesan cheese
2 tablespoons fresh lemon juice
4 large skinless boneless chicken breast halves
Cook frozen edamame in large saucepan of boiling salted water until just tender, about 10 minutes. Drain, reserving 1 cup cooking liquid for skordalia.
Meanwhile, combine basil, 1/3 cup oil, and garlic in processor; blend until basil and garlic are finely chopped. Transfer 2 tablespoons basil oil to small bowl and reserve.
Add edamame, 1/2 cup reserved cooking liquid, Parmesan, and lemon juice to remaining basil oil in processor; puree until mixture is almost smooth, adding more cooking liquid by tablespoonfuls if mixture is dry. Season generously with salt and pepper. Set skordalia aside.
Prepare barbecue (medium-high heat). Brush grill with oil. Brush chicken breasts on both sides with reserved basil oil; sprinkle with salt and pepper. Grill until chicken is just cooked through, about 6 minutes per side. Slice chicken breasts crosswise.
Spoon warm or room-temperature skordalia onto 4 plates. Top with chicken