In a small bowl, soak the shiitake mushrooms in boiling water for 30 minutes. Drain, snip off and discard the stems, and halve the caps. In a wok or saucepan, bring a few cups of generously salted water to a boil. Add the tofu slices and let soak for 3 to 5 minutes, then remove and blot dry with a clean tea towel.

Combine the starch with 1 tablespoon cold water in a small bowl. Heat ½ cup oil in a wok or skillet, or enough to cover the pan’s bottom by about ½ inch. When the oil is hot and shimmering, slide in as many tofu pieces as will fit in a single layer on the surface. Don’t move the tofu until a golden crust develops on one side, about 4 minutes. Flip and cook on the other side for about 3 minutes, transfer to a paper towel–lined plate, and repeat until all the tofu is golden brown, crispy, and slightly puffy. Transfer the frying oil to a heatproof container for another use, reserving 1 tablespoon in the wok.

Return the wok to medium-high heat. Stir-fry the shiitake mushrooms, garlic, ginger, and scallion whites until fragrant, about 2 minutes. Push the aromatics to one side of the wok, then add the chili bean paste and stir-fry briefly to release its flavor. Add the soy sauce, stock, and sugar. When the liquid is bubbling, add the fried tofu and carrot. Cover the wok and cook gently for 3 to 4 minutes to allow the tofu to absorb the flavors.

Uncover and add the snow peas. Return the heat to high and cook, stirring, until the liquid has reduced. Taste and adjust the flavors, adding more salt as needed. Give the starch slurry a stir and add it gradually to the wok, stirring continuously for 30 seconds, until the sauce is lightly thickened and clings to the tofu. Remove from the heat and stir in the scallion greens. Transfer to a dish and serve.