- Heat grill to high. Char eggplant, turning often, until cooked through and soft, about 30 minutes. Let cool slightly.
- Cut eggplant in half, and scrape flesh into a fine sieve set over a bowl; discard skin. Let stand for 2 hours to drain. Coarsely chop eggplant.
- Stir 1/2 cup eggplant into beef; reserve remainder for another use. Divide mixture into 4 equal pieces. Shape each into a 3 1/2-inch patty (about 3/4 inch thick).
- Reheat grill to high. Grill buns, cut sides down. Brush both sides of burgers with oil, and season with salt. Grill until charred, about 4 minutes per side for medium (3 minutes per side if using a grill pan).
- Meanwhile, mix together ketchup, pickles, yogurt, and mustard. Spread some sauce onto each bun half. Top bottom buns with lettuce, tomato, then burgers. Shave cheese onto burgers using a Microplane zester. Sandwich with remaining bun halves.
Drained eggplant can be refrigerated for up to 3 days.
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