Peel eggplant and slice into 3 x 1¼ inch strips. Place eggplant on a rimmed platter and sprinkle lightly with salt. Cover with a paper towel or clean cloth. Press with a weight (such as a pot filled with water) for several hours or overnight. Drain.
In a skillet, sauté eggplant in the vegetable oil over low heat until tender. Add vinegar and ½ teaspoon of the sugar. Cook 1 more minute. Drain on paper towels.
In a bowl, combine remaining ½ teaspoon sugar with sour cream, lemon juice, milk or buttermilk, and onion. Stir to mix. Add eggplant and toss to coat well. Cover and refrigerate up to 4 hours before to let flavors meld together.
Serve on appetizer-size pumpernickel or rye bread. Another classic option is to cut the eggplant and onion into small pieces and serve on slices of cooled, cooked potato. Garnish with fresh dill, if you like.