Heat olive oil in a large skillet over medium-high heat. Add chicken; cook until browned and no longer pink in the center, about 5 minutes. Add sweet potatoes and onion; cook and stir until onion is translucent, 2 to 3 minutes. Add eggplant, garlic, ginger, and turmeric; cook until fragrant, about 1 minute more. Pour in broth and simmer stew until thickened, stirring occasionally, about 20 minutes.