Step 1

Preheat oven to 425°F. Lightly coat a rimmed baking sheet with cooking spray. Stir together lamb, cheese, ¼ cup onion, 1 tablespoon oil, 1 teaspoon salt, and ½ teaspoon pepper in a bowl until just combined; shape into 4 patties.

Step 2

Combine panko and remaining 1½ teaspoons salt in a large ziplock plastic bag. Whisk egg whites in a large bowl until foamy. Dip eggplant wedges, 1 at a time, in egg whites and transfer to bag with panko. Once all eggplant has been added to bag, seal and shake well to coat. Arrange eggplant in an even layer on prepared baking sheet and coat generously with cooking spray. Bake until golden brown, about 20 minutes, flipping halfway through.

Step 3

Meanwhile, heat remaining ½ tablespoon oil in a large nonstick skillet over medium-high. Add lamb patties and cook, flipping once, until browned, about 4 minutes per side for medium.

Step 4

Place cucumber and remaining ¼ cup onion on a paper towel. Squeeze gently to release liquid. Transfer to a small bowl and stir in yogurt, lemon juice, and remaining ¼ teaspoon pepper.

Step 5

Place patties on buns with lettuce and yogurt sauce. Serve with eggplant fries and remaining yogurt sauce.