Stripe the eggplant with a vegetable peeler and cut into 1/2-inch-thick slices. Sprinkle the eggplant slices generously with salt, place in a colander and let sit 30 minutes. Rinse the eggplant under cold water and pat thoroughly dry.
Meanwhile, combine the olive oil and garlic in a small glass cup, and microwave just until the oil is hot and bubbling, about 20 seconds. Whisk together the yogurt, mint, cumin, cayenne, 2 tablespoons water and salt to taste in a small bowl.
Heat a grill pan over medium-high heat or prepare an outdoor grill for medium-heat cooking. Brush the eggplant slices on both sides with the oil and garlic mixture. Grill until the eggplant is browned and tender, moving and turning the slices frequently, about 15 minutes. Drizzle the mint sauce onto the eggplant slices.
When grilling the eggplant, watch carefully to make sure that the slices cook evenly. Moving them frequently will help you avoid scorching them on the hotter parts of the grill before they cook all the way through. If you're using a grill pan, you'll want to cook the slices in 2 batches to avoid crowding the pan.
Recipe courtesy Food Network Kitchens