You will need 16 to 20 small or medium wooden skewers. Soak them in water for 20 minutes before using.
Meanwhile, peel and devein the shrimp, leaving the tails on if you prefer to eat the shrimp by hand. Pat them dry with paper towels and place them in a quart-size resealable plastic food storage bag.
Mince the garlic and add it to the bag. Finely chop the mint leaves, reserving a third of the chopped leaves for garnish; add the rest of the mint to the bag. Season with salt and pepper to taste. Drizzle in the oil as needed. Seal the bag and massage the contents to coat the shrimp evenly. Refrigerate for 10 minutes (or up to 30 minutes for deeper flavor).
Position an oven rack 6 to 8 inches from the broiler element; preheat the broiler. Have a rimmed baking sheet at hand.
Grate the cheese (to yield 1/4 cup) into a wide, shallow bowl, then add the breadcrumbs. Shake off any excess marinade from the shrimp, then press them into the cheese-breadcrumb mixture to coat them on both sides.
Thread 4 or 5 shrimp onto each skewer, laying the soaked skewers flat on the baking sheet as you work. Broil for 2 minutes on each side, so the coating is golden and the shrimp turn pink. Be careful not to overcook.
Arrange the skewers on individual plates; top with any stray breadcrumbs from the baking sheet. Cut lemon wedges and squeeze a few over each skewer and sprinkle with the remaining chopped mint. Serve hot, with the remaining wedges passed at the table.