Make a pocket-like slit in each eggplant, trying to not cut to the edges.
Place eggplants in a large stock pot filled with water. Bring to boil, and cook on low until the skin is just tender, 8-10 minutes.
Once eggplants are cooked, drain them through a sieve and leave in a sieve with a weigh on top (a plate with a jar of water of top) - for about an hour - to remove moisture.
Prepare the filling: shred carrots with a mandolin slicer or a grater. Remove the tough stems from parsley (reserve some for tying eggplants later). Press or chop garlic. Using rubber gloves, finely chop chili pepper removing the seeds for less heat (from experience, if you don't use gloves, your hands will burn for quite a while afterwards). Mix everything together.
Stuff the eggplants with the filling.
Make brine: Bring water to boil in a small pot, add the rest of the ingredients; remove from heat as soon as salt is dissolved. You can toss the parsley stems in hot water for a few minutes to make parsley more flexible for tying. Cool the brine.
Tie stuffed eggplants with parsley, place in fermentation crock, add the brine.
Place a weight on top, and cover tightly.
Leave at room temperature for 3-5 days, then move to refrigerator (I transfer them to a mason jar). They should keep for up to a year but I haven't gone longer than 2 months, we eat them fast!
Notes
Don't be shy with parsley, you can't have too much here, the more the better :)