In a skillet, heat the oil until shimmering. Add the shiitake, scallions, bok choy and carrot; stir-fry over high heat until tender, 5 minutes. Add the garlic, ginger, soy sauce, mirin and chile-garlic sauce and cook until the liquid is evaporated, 3 minutes. Season with salt. Transfer to a bowl and refrigerate until chilled, 15 minutes.
Line a large baking sheet with wax paper. Add the pork to the filling and mix. On a work surface, brush 3 wonton wrappers with the cornstarch solution; spoon a scant tablespoon of the filling in the center. Fold the wrappers over to form triangles; seal the edges, pressing out any air trapped inside. With scissors, trim the the wrappers, leaving a 1/4-inch rim around the filling. Transfer the dumplings to the baking sheet, seam sides up. Keep the dumplings covered with a moist paper towel. Repeat with the remaining wonton wrappers and filling.
Combine all of the ingredients in a medium bowl.
Fill a wok or a large skillet with 2 inches of water and bring to a boil. Working in batches, arrange the dumplings in a double-tiered bamboo steamer lined with oiled wax paper; set the steamer over the boiling water. Cover and steam the dumplings until the filling is cooked through and firm, 6 minutes.
Toss the dumplings in 1 cup of the sauce. Using a slotted spoon, transfer the dumplings to a platter. Serve the remaining chile-sesame sauce alongside.