Prepare a medium-high gas or charcoal grill fire.
In a small bowl, whisk 3 Tbs. of the peanut oil with the lime juice, 2 Tbs. of the shallot, the fish sauce, 3/4 tsp. of the sugar, and the chiles. Season to taste with salt and pepper.
In another small bowl, combine 2 tsp. water with the ginger, soy sauce, the remaining 1 Tbs. shallot, and 1 tsp. sugar.
Arrange the eggplant halves on a rimmed baking sheet, brush both sides with the remaining 1 Tbs. peanut oil, and sprinkle with salt and pepper. Grill the eggplant, covered, until tender, 3 to 5 minutes per side.
Combine the lettuces, tomatoes, basil, and mint in a large bowl. Rewhisk the lime dressing and toss just enough into the salad to lightly coat the greens. Season the salad to taste with salt and pepper. Transfer the salad to a platter and arrange the eggplant over the salad. Spoon the ginger mixture over the eggplant, and serve immediately.