Combine Greek yogurt, chopped cucumber, mint, lemon juice, lemon zest, and grated garlic in a medium bowl.

Cover and refrigerate cucumber sauce.

Place lamb shoulder pieces on a parchment-lined baking sheet and freeze for 20 minutes.

Attach a meat grinder to a stand mixer and grind the lamb with 2 cloves of garlic.

Remove the grinder and attach a flat beater to the mixer.

Add red onion, mint, parsley, dried red chilies, cumin, salt, and pepper to the ground lamb mixture.

Mix until combined, about 30 seconds.

Shape the mixture into 6 patties.

Heat 1/2 tablespoon of olive oil in a large skillet over medium-high heat.

Cook halloumi slices in batches for 6-8 minutes, or until browned on both sides.

Remove halloumi from the skillet and set aside.

Add 1/2 tablespoon of oil to the same skillet and heat over medium heat.

Cook burgers in batches until cooked through, about 3-4 minutes per side, adding remaining oil as needed.

Serve the burgers immediately with grilled halloumi and yogurt sauce on toasted buns.