Cut eggplant crosswise into 1/2-in. slices. Put slices in salted water; top with a plate or pot lid to submerge.
Let sit 30 minutes.
Meanwhile, prepare a charcoal or gas grill for medium heat (350 to 450; you can hold your hand 5 in. above cooking grate only 5 to 7 seconds).
Drain eggplant on a kitchen towel. Arrange eggplant and tomatoes on a baking sheet.
Brush all over with 1/2 cup oil.
Grill tomatoes, covered, turning once, until grill marks appear, 2 to 3 minutes per side.
Transfer to a plate and set aside.
Grill eggplant (in batches if necessary), covered, turning occasionally, until browned all over and tender to the bite, 5 to 8 minutes. Arrange on a large platter.
In a blender or food processor, pure remaining 1/2 cup olive oil, the lemon juice, 1/2 cup mint leaves, the garlic, remaining 1/2 tsp. salt, the pepper, and mustard.
Drizzle three-fourths of dressing over eggplant. Arrange tomatoes on eggplant, drizzle them with remaining dressing, and top with remaining 1/2 cup mint. Season to taste with more salt.
Make ahead: Prepare recipe through step 5 up to 3 hours ahead, cover, and hold at room temperature. Assemble just before serving.
