Part lean beef, part eggplant, this low-calorie burger offers the best of all worlds. Shaving extra-sharp cheddar cheese on top makes a small amount go a long way. For the best texture, cook the burgers to medium.
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.