Clean and trim the scallions. Cut off the green parts from 4 of the scallions and reserve for garnish. Cut all of the remaining scallions into thirds.

Cut the 4 scallion greens in half cross-wise. Using a sharp knife, cut the halved pieces in half again lengthwise to create a flat side. Very thinly slice each of these flat pieces of scallion greens into thin matchsticks, about 1/8-inch or 1/16-inch wide. The scallion slices will curl naturally. Set them aside in a bowl and reserve for garnish.

Heat the oil in a wide pot over medium heat. Add cut scallions to the pot and cook, stirring occasionally, until bright green about 5–7 minutes. The scallions shouldn’t brown, so adjust the heat as necessary to prevent browning.

Add the diced potato and pour in the water. Season liberally with salt and pepper. Bring to a boil and then reduce heat and simmer for 25 minutes or until the potato is fork tender.

As the broth simmers, bring a large pot of water to a boil. Cook the udon according to package instructions and then drain and rinse. Set aside.

After the broth has simmered, add the spinach to the pot and cook for 5 minutes until bright green. Do not overcook the spinach! Turn off the heat and allow the broth to cool for 5 minutes.

Transfer the broth to a food processor or blender. Pulse until smooth. Add the coconut milk and blend once more until smooth. Taste and season with salt and pepper.

Ladle the broth into shallow bowls. Place a portion of the udon on top and arrange the scallion slices on top. Garnish with a drizzle of sesame oil and a sprinkle of black sesame seeds. Enjoy!