Preheat gas grill (or prepare charcoal fire) to medium-high heat; brush and lightly oil grates.
Combine cucumber, watermelon, carrots, mint, lemon juice, zest, and vinegar in a small bowl; toss to coat.
In another small bowl, mix together the crumbled feta, yogurt, mayo, and 1 teaspoon of the Greek seasoning.
For the patties, place the lamb, turkey, onion, and 1 tablespoon remaining Greek seasoning in a large bowl and use hands to gently mix together. Shape into 6 even patties to fit roll size. Place butter pieces and hot wing sauce in a microwave-able bowl and microwave on high until butter is melted, about 45 seconds. Stir/whisk until well incorporated and set aside 1/4 cup of the glaze for drizzling. Liberally brush tops of patties with some of the glaze, place on grill, glazed side down. Brush patty tops with some more glaze.
Cook 12-14 minutes, turning once, or until cooked through.
Spray cut sides of rolls with some olive oil cooking spray and lightly toast on outsides of grill.
Place a patty on bottom roll half, top with rounded tablespoonful of the feta spread, top with some cucumber slaw, allowing it to spill over and around burgers, and drizzle tops with some of the reserved glaze. Close with roll tops and serve, with extra napkins.