1 Line A Large Tray With Paper Towels. Lay Out The Rounds Of Eggplant On The Tray In A Single Layer. Salt Generously On Both Sides. Let Sit While You Prepare The Other Ingredients. 2 Cut The Tomato Into 1/4 Inch Thick Slices. Cut The Red Onion Into 1/8 Inch Slices. Pour The Two Vinegars And A Teaspoon Of Olive Oil Into A Flat Container Such As A Casserole Dish. Sprinkle With A Little Salt And Stir With A Fork To Combine. Lay The Onions And Tomatoes Down In The Container And Turn To Coat With The Dressing. 3 Prepare Your Grill For High, Direct Heat. Make Sure The Grates Are Clean, And Close The Lid. 4 Once The Eggplant Slices Have Been Sitting For At Least 15 Minutes, Pat Them Dry With More Paper Towels. Use A Pastry Brush To Paint Them On Both Sides With Olive Oil. Lay The Slices On The Grill And Grill Them For About 3-4 Minutes Per Side With The Lid Closed, Until Nicely Browned. Remove From Grill. Lightly Toast The Burger Buns On The Grill, For About 30 Seconds To A Minute, Just Enough To Get Lightly Toasted. 5 To Build The Sandwich, Set Out Your Burger Buns. Put Two Or Three Rounds Of Eggplant Slices On Top Of The Bottom Bun. Then Add A Tomato Slice And Some Red Onion. Add A Couple Slice Of The Fresh Mozzarella And Top It With Some Basil Leaves.