Using a vegetable peeler, remove lengthwise strips of skin from each of the eggplants at 1-inch intervals. Slice each eggplant crosswise into 1/2-inch-thick rounds (you should have 15 to 20 rounds). Arrange the eggplant on two paper towel/ lined baking sheets in a single layer. Rub the eggplant slices with the kosher salt and let stand until juices form, about 1 hour. Pat with paper towels to remove excess liquid.
Meanwhile, in a medium bowl, combine the tomatoes, 3 tbsp. EVOO, mint, garlic, vinegar, crushed red pepper, cumin and cayenne; season with salt. Let stand until flavors meld, about 2 hours.
Preheat a grill to medium-high. Brush the eggplant slices with EVOO. Working in batches, if needed, grill the eggplant until charred and tender, 4 to 6 minutes per side. Arrange on a platter in a single layer.
Stir the cheese into the tomato mixture. Mound onto the eggplant slices and drizzle with EVOO.