Step 1: Choose the Blade
Put aside the julienne blades that cut foods into matchsticks; you want one of the slicing blades that cuts into slices or planks. Choose the thickness level you want -- depending on your model, you may have paper-thin settings up to about 1/4 inch. For eggplant, the upper thickness blades or settings are optimal. If you're preparing eggplant for a rollatini -- go with about a 1/8-inch thickness; for Parmesan or planks to fry or grill, opt for 1/4 inch.
Step 2: Use Safely
Plan to use the hand guard or a blade-safe glove when slicing. Every mandoline model is slightly different, but usually the hand guard features "teeth" that insert into the vegetable so you can run it up and down the blade with your hand safely out of harm's way.
Step 3: Prepare the Eggplant
Rounds are probably the easiest to cut with most mandolines. Peel the eggplant before slicing if you want to avoid the skin in your dish -- it can be tough and bitter in older eggplants. Trim the top stem portion of the eggplant off using a knife and insert the blade protector in the fatter end. Place the trimmed end of the eggplant at the top of the slice and apply slight pressure as you run the vegetable over the blade. For planks, trim off both ends of the eggplant, insert the blade protector and run the vegetable lengthwise over the blade.
Use even pressure as you slice the eggplant; otherwise, you'll end up with slices of different thickness. Some blame uneven slices on the mandoline's quality, but it's usually a result of user error.
Step 4: Discard
If you've made planks, discard the first and final slice, which are mostly skin. If you've made rounds -- all except for those at the very end are usable.
For your first time using a mandoline, buy a few extra eggplants to practice slicing. You really have to get a feel for your model and the vegetable. You could also use potatoes or zucchini for practice, but know that they'll have a slightly different feel than the eggplant.