Trim the chicken and cut it into bite-size chunks. Trim the eggplant, cutting away stems and bottoms. Quarter each lengthwise, and then slice crosswise at 1-inch intervals.
Heat the oil in a 2- to 3-quart saucepan over medium heat until a bit of curry paste just sizzles when added to the pan. Add all the curry paste and cook, pressing and stirring with a wooden spoon or heatproof spatula to soften the paste and mix it in with the oil, until fragrant, about 2 minutes.
Increase the heat to medium high and add the chicken. Spread it in a single layer and cook undisturbed until it starts to brown around the edges, 1 to 2 minutes. Stir and continue cooking until most of the chicken is lightly browned, 1 to 2 minutes.
Add the coconut milk and chicken broth and stir well, scraping the bottom of the pan to release the browned bits. Add the eggplant and half the lime leaves (if using) and bring to a simmer.
Simmer, adjusting the heat as needed and stirring occasionally, until the chicken is completely done and the eggplant is tender, 8 to 10 minutes.
Add the fish sauce, sugar, and salt and stir. Remove from the heat. Tear the basil leaves in half (or quarters if they are large), and stir them into the curry, along with the remaining lime leaves (if using). Let rest for 5 minutes to allow the flavors to develop.
Serve hot or warm with rice or noodles, garnished with the chile slices (if using).