Whisk together soy sauce, vinegar, brown sugar, and 3 tablespoons oil; season with salt. Place chicken in a resealable plastic bag with 1/4 cup soy-sauce mixture. Cook noodles in a large pot of boiling water according to package instructions until al dente; add edamame for last minute. Drain, rinse under cold water, and toss with 1/2 cup of remaining soy-sauce mixture.
Heat grill to medium-high. Brush eggplant with remaining oil; season with salt and pepper. Remove chicken from marinade. Grill chicken and eggplant, flipping once and brushing both sides with remaining 1/4 cup soy-sauce mixture, about 3 minutes for chicken, 7 minutes for eggplant. Transfer to a plate; cut chicken into large slices. Top noodles with chicken, eggplant, and some juices from plate. Serve with mint, scallions, and sesame seeds.