Place the ginger, shallot, garlic, turmeric, lemongrass, chilies and coriander in a food processor with 3 tablespoons of coconut milk. Puree until finely blended.
In a large pot over medium heat, add this mixture and cook for 3–5 minutes. Add the lime zest/Kaffir lime leaves, lime juice, star anise, brown sugar, soy sauce and salt. Continue to cook for another 1–2 minutes, until the sauce becomes fragrant. Add the remaining coconut milk, mizuna greens, potato, carrots, chicken broth, water and the fish sauce. Continue to simmer over medium heat until the carrots and potatoes are tender. Taste the soup and adjust seasonings. There should be a nice balance of salty, sweet, sour and spicy heat. Serve while hot.
Note: 1 cup of cooked and sliced chicken or tofu can be added if serving as a main course.
* Can substitute ¼ to ½ teaspoon dried turmeric.