Preheat the oven to 325 degrees F.
In a large ovenpoof skillet over medium-high heat, add all of the marinade ingredients. Bring to a boil and cook until reduced by half. Add the eggplant and mushrooms and cook for 5 minutes.
Put the pan into the oven and cook until the eggplant and mushrooms are soft and caramelized, about 4 to 5 minutes. Transfer to a serving bowl and garnish with sesame seeds.