Soak wooden skewers in water for at least 15 minutes.
Preheat grill for medium heat and lightly oil the grate.
Chop garlic on a cutting board. Sprinkle kosher salt over salt and with the back of a large knife, smash the garlic on the cutting board to form a paste.
Heat garlic paste and olive oil in a skillet over medium-low heat until garlic begins to brown, about 5 minutes. Remove from heat.
Skewer about 5 shrimp on each wooden skewer by piercing shrimp through the tail and the upper body. Season shrimp with kosher salt and pepper. Brush one side of the shrimp with garlic-infused olive oil.
Place shrimp oil-side down on the preheated grill and brush with more olive oil; cook until shrimp begins to turn pink and curl, about 4 minutes. Turn shrimp over and brush with olive oil again. Continue grilling until shrimp is opaque and pink all over, about 4 minutes more.