In a small bowl, mash feta with the back of a fork until smooth. Stir in yogurt and season with salt and pepper. Set aside.
In small heatproof bowl, combine bulgur and 1/2 cup boiling water. Cover with plastic wrap and let sit 10 minutes. Meanwhile, in a food processor, pulse chickpeas until coarsely chopped. Transfer to a large bowl and add beef and mint. Drain bulgur and add to chickpea mixture. With your hands, mix until mixture is evenly combined and holds together when pressed into a ball. Divide into four 1-inch-thick patties and season with salt and pepper.
Lightly coat a large nonstick skillet with cooking spray and heat over medium-high. Add patties and cook until browned and crispy on the outside and cooked through, 8 to 10 minutes, flipping once. Serve on buns with sliced cucumber, tomato, and feta-yogurt sauce.
Unlike most whole grains, you barely need to cook bulgur -- simply cover it with boiling water and let sit for 10 minutes.