Heat a grill to medium (350° to 450°). Make hummus: Whirl garlic, tahini, oil, lemon juice, cumin, salt, and edamame in a food processor until mostly smooth.
Make slaw: In a medium bowl, toss carrots with lemon juice, salt, and pepper; let stand 15 minutes, stirring occasionally.
Make burgers: Brush veggie patties with oil and grill, turning once, until grill marks appear, about 6 minutes.
Arrange 1 lettuce leaf on each of 4 plates. Top each with a veggie burger patty and 2 tbsp. hummus. Divide slaw, tomato, cucumber, and sprouts between burgers and top each with another lettuce leaf.
*Available at well-stocked grocery stores.
Note: Nutritional analysis is per burger.