Heat the vegetable oil in a large skillet over medium-high heat until shimmering. Add the mushrooms and cook until they have released their moisture and turned golden brown, about 5 to 7 minutes.

Stir in the garlic and ginger and cook until fragrant. Transfer the mixture to a medium bowl and allow it to cool slightly before stirring in the tofu, kimchi, scallions, soy sauce, and sesame oil. Mash the mixture with a fork until it is cohesive enough to hold together.

Place a wrapper in the palm of your hand and spoon a tablespoon of filling into the center. Bring the sides of the wrapper up around the filling, pleating as you go, to create an open-faced cylinder. Gently squeeze the center to secure the wrapper and tap the bottom of the dumpling on a flat surface so it stands upright.

Line a steamer basket with the cabbage leaves to prevent sticking. Arrange the dumplings in the basket, leaving space between them so they do not touch. Steam over boiling water until the wrappers are translucent and the filling is piping hot, about 6 to 8 minutes.

Whisk the soy sauce, rice vinegar, and chili oil in a small bowl. Serve the dumplings immediately while they are hot and tender.