Spread the tofu over a paper towel–lined baking sheet and let it drain for 20 minutes, then gently pat it dry with fresh paper towels. Squeeze as much excess liquid as possible from the kimchi to prevent the filling from becoming soggy.

Pulse the tofu and kimchi in a food processor until coarsely chopped, about 5 pulses—the pieces should be no larger than a quarter-inch. Transfer the mixture to a large bowl and stir in the mushrooms, soy sauce, cornstarch, cilantro, sesame oil, wine, vinegar, sugar, ginger, salt, and pepper until the mixture is well combined and slightly tacky.

Cut two 3-inch rounds from each wonton wrapper using a biscuit cutter to create roughly 40 rounds total. Keep the rounds covered with a moist paper towel while you work so they stay pliable.

Brush the edges of a wrapper round with water and place a heaping tablespoon of filling in the center. Pinch the wrapper around the sides of the filling, leaving the top exposed, then use the flat side of a butter knife to pack the filling down firmly and create a flat top. Top each dumpling with a small pinch of grated carrot.

Line a steamer basket with a piece of parchment paper cut slightly smaller than the diameter, poke about 20 small holes in the paper, and lightly coat it with vegetable oil spray. Arrange the dumplings in the basket so they do not touch and steam over simmering water until the wrappers are tender and translucent, about 6 minutes.

Serve immediately while steaming hot, accompanied by chili oil for dipping.