Peel garlic cloves and slice them very thin. Thinly slice scallions on the bias, reserving the green parts for serving. Peel ginger and slice into very thin strips. Peel and julienne carrots. Remove stems of shiitake mushrooms and slice the caps thinly. Pull the fibrous strings from the snow peas and slice the peas thinly on the bias.

Warm sesame oil in a large pot over medium heat. Add garlic, the white parts of the scallions, and ginger. Sweat the aromatics until they start to soften, about 3 minutes. Add broth, soy sauce, and mirin, and simmer broth for 10 minutes. Remove the pan from the heat. Dissolve miso in 1/2 cup of broth and add the mixture back into the pot along with carrots, mushrooms, snow peas, and cabbage. Cover and let the vegetables cook in the residual heat for 2–3 minutes.

Divide soup among 4 bowls to serve along with freshly boiled dumplings. Garnish with chopped scallion tops and a swirl of chile oil if using.