In a bowl, whisk the miso, garlic, sesame oil and 1/2 cup of the stock. In a large skillet, heat the vegetable oil. Add the chicken, season with salt and pepper and cook over high heat until just cooked through, 4 minutes; using a slotted spoon, transfer to a plate.
Add the shiitake, ginger and the remaining stock to the skillet. Cover and cook over moderate heat until the shiitake are tender, 2 minutes. Add the cabbage and chicken and stir-fry over high heat for 1 minute. Add the miso sauce and bring to a simmer. Stir in the daikon and scallions. Transfer to bowls and serve.