Slice 1 eggplant lengthwise into thirds; toss with 2 T. EVOO and 1/2 t. kosher salt. Grill over medium-high heat until tender, 8 mins. per side; coarsely chop. Toss with 1 chopped tomato, 1 minced garlic clove, 2 T. chopped basil, and 1 t. each red wine vinegar and chopped oregano. Season with salt and pepper and top with shaved parmesan.
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