Grab a wok or a large skillet. Add the thick coconut milk (the stuff floating on the top of the can) and bring to a lively simmer. Let it go until the cream starts to break. You will see oil separating. This oil is what you will fry the curry paste in.
This doesn't always work. At least it doesn't always work for me. I think it has to do with how thick the coconut cream is. Sometimes it's almost solid. Other times its more like whipping cream. It's not a big deal in this recipe because there is already some oil in the paste. I usually give up after 6 or 7 minutes. If it hasn't split by that point give up.
Once you see the oil or give up add the curry paste and stir to combine. Cook, stirring, until the paste loses it's pungent raw smell. It will smell mostly of the shallots at first. You are done when that smell is gone. This takes about 5 minutes.
Mix in the chicken and cook until the chicken is coloured. Add the remaining coconut milk. Simmer 5 minutes.
Add the quartered Thai eggplant, kaffir lime leaf and fish sauce. Cook until the chicken is done, another 15 minutes or so.
Add the basil and cook one minute more. Taste for salt. You may need a little more. You may not. Up to you.
Serve garnished as you like. Fried shallots, more chilies, basil garnish and lime all work.