Wake up the paste: heat the thick coconut cream from the top of the can in a large wok or Dutch oven over medium-high heat. Add the curry paste and oil; fry 2-3 minutes, pressing against the hot surface, until deeply fragrant and a shade darker.
Add the chicken pieces and stir to coat in the paste. Cook 4-5 minutes, stirring now and then, until opaque on the outside (it does not need to be cooked through yet).
Build the sauce: pour in the remaining coconut milk, broth, fish sauce, sugar, and kaffir lime leaves. Bring to a gentle simmer, lower the heat, and bubble softly for 8 minutes. Taste and adjust fish sauce and sugar until savory, gently sweet, and warmly spicy.
Add the Thai eggplant, bamboo shoots, and red bell pepper. Simmer 8-10 minutes more until the eggplant is fork-tender and the chicken is cooked through. Add a splash of water or broth if the sauce reduces too much.
Finish: take off the heat and stir in the Thai basil leaves until just wilted. Serve over hot jasmine rice with lime wedges, extra basil, and sliced chiles on the side.