Wash the eggplant.  I peeled mine per Brian's request, but the original recipe leaves the skin on, hence the "black-blue" in the original title.  Cut the eggplant into slices about 2" by 4" by 1/2" (this can be done by cutting the eggplant in half width-wise and slicing from there.  Mine were pretty randomly sized, actually)

Salt the eggplant (this is exactly what it sounds like: sprinkle salt on the eggplant.  Don't be embarrassed, I had to look it up too).  Place the eggplant in a colander for about half an hour.  This removes the excess liquid from the eggplant.

Preheat oven to 375°.

Saute the onions in 2 tablespoons of oil for 7-10 minutes, until the onions are soft, but not brown.  Place them on a plate.

Use paper towels to pat the eggplant dry.  Brown the eggplant in oil several pieces at a time using the pan from the onions.

Lightly oil a 8x8 baking dish.  Put half the onions and 1 tsp dried basil on the bottom of the dish.

Layer the eggplant, tomato and cheese slices vertically in the pan, creating visible stripes.

When the dish is full (and hopefully you're out of ingredients), cover the stripes with the remainder of the onions and basil.

Cover the dish with foil and bake for 40 minutes.