Place chickpeas in food processor and pulse until finely chopped. Transfer chickpeas to medium bowl; stir in couscous, carrot, feta, 2 tablespoons dill, olives, egg white, and salt, stirring until mixture clumps together. Shape mixture into 6 (3-inch) patties. Spray patties with nonstick spray.
Spray large ridged grill pan with nonstick spray and place over medium-high heat. Place patties in pan and grill until golden brown and heated through, 3−4 minutes per side.
Meanwhile, scrape away gills from under mushroom caps with spoon. Mix yogurt and remaining 1 tablespoon dill in small bowl.
Place 1 chickpea patty in each upturned mushroom cap, gently pressing each to fill caps. Return burgers to one side of grill pan, mushroom side down. Spray onion slices with nonstick spray and add to other side of pan. Grill burgers until mushrooms are tender and onions are lightly charred, about 8 minutes, turning onions halfway through grilling time.
Serve each burger topped with 1 lettuce leaf, 1 tomato slice, 1 grilled onion slice, and 1 tablespoon yogurt-dill sauce.
Serving size: 1 burger
Notes
You can also use cannellini beans in place of chickpeas for this recipe. The burgers are also freezable, making them a great make-ahead option for busy weeknights.