Preheat oven to 400. Arrange chicken on two rimmed baking sheets. Rub with 1 tablespoon oil; season with salt and pepper. Sprinkle 4 breasts with jerk seasoning; roast all until an instant-read thermometer inserted in the thickest part of breast (avoiding bone) registers 165, 40 to 45 minutes.
Meanwhile, heat 4 tablespoons oil in a large skillet over medium-low. Add half the plantains, and cook until browned, 4 to 6 minutes per side. Transfer to a paper-towel-lined plate, and season with salt. Repeat with remaining plantains. Set aside.
Make salad: In a small bowl, whisk together vinegar, mustard, orange juice, and remaining 2 tablespoons oil; season with salt and pepper. In a large bowl, combine lettuce and orange segments. Drizzle with dressing, and toss to combine. Serve chicken with salad and plantains.
Pick plantains that look like their overripe banana cousins for the best flavor. The darker the skin, the sweeter the plantain.