Peel the potatoes and cut them into ½-inch cubes. Wash the leeks well. Trim and discard the dark green parts. Cut them in half down the center, checking for any remaining dirt. Slice the leeks into thin half rings.
Heat the olive oil in a large saucepan on medium flame. Add the vegetables and cook and stir them for several minutes, until the leeks have softened. Add the salt and pepper.
Stir in the white wine. If you are including the white miso, add only 3 cups of the vegetable broth; otherwise add 4 cups of vegetable broth to the potatoes and leeks.
Cover the pot. Turn down the heat so the liquid stays at a simmer and cook until the potatoes are very tender, 15 to 20 minutes. Alternatively, use a pressure cooker or Instant Pot and cook under pressure 5 minutes. Use the quick release method. (See note.)
Puree the hot liquid, potatoes and leeks directlly in the cooking pot using an immersion blender. If you don't have an immersion blender, puree the broth and vegetables in a blender, about 2 cups at time, with caution.
To finish the soup, dissolve the miso in 1 cup hot water and stir into the pureed potatoes and broth. Season to taste with more salt and white pepper.
Refrigerate, or stir over an ice bath until very cold. Serve garnished with the avocado and/or chives.