Cut tough stems from spinach; submerge in cold water and rinse to remove dirt.
Heat oil in a large straight-sided skillet over medium-high heat. Add onion and cook, stirring often, until softened, about 5 min. Stir in coconut milk, broth, curry paste, and salt; bring to boil, reduce heat and simmer 8 min.
Add chicken and eggplant, bring to low boil, then reduce heat to simmer and cook, stirring occasionally until chicken is done and eggplant is tender, about 8 min. Add spinach, stir, and cook 1 min more. Stir in lime juice and serve.