Place hoisin sauce and peanut butter in a small bowl and stir well to combine. Set aside.
In another small bowl, whisk the egg with 1 tsp. water and a pinch of salt. Heat 1tsp. oil in a wok or large frying pan over medium heat until shimmering, about 2 minutes. Add the egg and quickly, with a twist of the wrist, swirl the pan to coat it with the thinnest layer you can. When the egg is set but not browned on the bottom, flip and cook until the other side is just set (this happens in seconds). Transfer to a cutting board and cut into 1⁄4-in.-wide ribbons.
In a small bowl, whisk the cornstarch with 1 tsp. water and set aside. Heat the remaining 1 tbsp. oil in the same pan over high heat until smoking. Add the mushrooms, season with salt and pepper, and, using a metal spoon or spatula, stir constantly until the mushrooms are browned and their liquid is evaporated, about 2 minutes. Add the mushroom broth, Shaoxing rice wine, soy sauce, vinegar, and sugar and bring to a boil. Add the cornstarch mixture to the juices at the bottom of the pan, and keep stirring until a brown sauce coats the mushrooms. Transfer the mushrooms and sauce to a shallow bowl or platter. Top with the egg ribbons and cilantro. Serve with pancakes spread with the hoisin sauce–peanut butter mixture.