Greek Lamb Burgers With Baked Eggplant Fries

Name: Greek Lamb Burgers With Baked Eggplant Fries
Ingredients: Nonstick cooking spray; 1 pound ground lamb; 2 ounces feta cheese, crumbled (about ½ cup); ½ cup grated red onion (from 1 small onion), divided; 1 ½ tablespoons olive oil, divided; 2 ½ teaspoons kosher salt, divided; ¾ teaspoon freshly ground black pepper, divided; 1 ½ cups panko; 2 large egg whites; 1 medium eggplant, cut into ½-by-1-by-2-in. wedges; ½ cup grated English cucumber (from ½ cucumber); 1 cup plain whole-milk Greek yogurt; 2 teaspoons fresh lemon juice (from 1 lemon); Hamburger buns and lettuce, for serving