A layered chopped salad mixes lemongrass-marinated chicken with tropical fruit and summer vegetables.
Place chicken between sheets of plastic wrap; pound to 1/2-inch thickness. Place in a plastic or glass dish; coat with marinade. Cover; chill 4 hours or overnight.
Heat grill or broiler. Season chicken with salt and pepper; grill until golden brown and cooked through, 4 to 5 minutes per side. Set aside to cool.
Fill a large bowl with ice and water; set aside. Bring a large saucepot of water to a boil; add salt. Add snap peas; cook until bright green and still crisp, about 2 minutes. Transfer to ice bath to cool. Drain; cut into 3/4-inch pieces.
Place shallots and ginger in a food processor; process until very finely chopped. Add lime juice, 1/2 cup mint, 1/2 cup cilantro, the salt, and pepper; process until herbs are finely chopped. Slowly add oil through feed tube with motor running.
Layer romaine, mangoes, chicken, red pepper, fennel, snap peas, and scallions in a large bowl. Top with remaining mint and cilantro. Serve with dressing on the side or add dressing to bowl.