Preheat a grill pan over medium heat. In a small bowl, mash three-quarters of the feta, the lemon peel and 1 1/2 teaspoons olive oil.
In a medium bowl, combine the lamb with 1/4 cup pomegranate seeds and the remaining feta. Shape into 4 patties. Grease the grill pan with 1 1/2 teaspoons olive oil, add the burgers and cook, turning once, for 8 minutes for medium. Transfer to a plate and cover loosely with foil.
Meanwhile, in a bowl, toss the arugula with the chopped mint, the remaining 1 tablespoon olive oil and the lemon juice; season with salt.
Spread the bun bottoms with the feta mixture. Press the remaining pomegranate seeds into the cheese. Top with the burgers, red onion, arugula salad, whole mint leaves, and bun tops.