Heat 1 tablespoon olive oil in a large, heavy-bottomed skillet over medium-high heat. Add eggplant and cook, stirring occasionally, until browned, about 5 minutes. Transfer to a bowl.
Heat another tablespoon of oil in the skillet. Add mushrooms and saute until tender, about 5 minutes. Transfer to the bowl with the eggplant.
Add remaining oil, onion, garlic, and Thai chile to the skillet. Cook until onion begins to soften, about 5 minutes. Stir in coconut milk, broth, curry paste, snow peas, bell pepper, and salt. Return eggplant and mushrooms to the skillet.
Bring curry to a simmer, reduce heat to low, and cook until vegetables are tender, about 15 minutes.
