Place each chicken breast half between two sheets of plastic wrap; with a flat mallet or rolling pin, gently pound to 1/2 inch thick. Brush both sides of chicken lightly with olive oil and sprinkle with salt and pepper.
Lay chicken on an oiled grill over a solid bed of hot coals or high heat on a gas grill (you can hold your hand at grill level only 2 to 3 seconds); close lid on gas grill. Cook, turning once, until chicken is no longer pink in the center (cut to test), 6 to 8 minutes total. Meanwhile, brush both sides of bread lightly with oil. When you turn chicken, lay bread slices on grill and cook, turning once, until lightly toasted, about 4 minutes total.
To assemble each sandwich, top one slice of bread with about 1/4 cup arugula leaves. Place chicken on arugula and top with about 1/3 cup fig relish. Top with second slice of grilled bread. Serve warm.