In a screw-top jar combine balsamic vinegar or red wine vinegar, the 1/4 cup olive oil, the mustard, and garlic; cover and shake well. Season to taste with sugar, salt, and pepper. Set aside.
Place fish in a shallow dish; set aside. Pour 1/4 cup of the vinaigrette over fish, turning to coat. Cover and marinate at room temperature for 20 minutes, turning fish once. Drain fish, discarding marinade.
Meanwhile, in a saucepan cook potatoes in a small amount of boiling salted water for 10 minutes, adding green beans for the last 3 minutes of cooking. Drain and cool slightly. Place potatoes and beans in a bowl and toss with olive oil. Spread vegetables in a single layer in a grill basket.
Lightly grease the rack of a gas grill. Preheat grill. Reduce heat to medium. Place fish and basket of vegetables on the grill rack directly over heat. Cover and grill until fish just flakes easily when tested with a fork and vegetables are tender, turning once halfway through grilling. (Allow 8 to 12 minutes for fish and about 10 minutes for vegetables.) Remove from grill. Cut fish diagonally into 1/4-inch slices.
In a large bowl toss together romaine, cherry tomatoes, olives, and red onion. Divide among 4 dinner plates. Arrange the fish, potatoes, beans, and eggs on top of romaine mixture. Drizzle with the remaining vinaigrette. Sprinkle with tarragon.