Toast the sesame seeds in a dry pan, stirring constantly, until they start to brown. Remove from the pan and set aside. Trim the ends and destring the sugar snap peas, then chop both the peas and pods together. (If using English peas, shell the peas and find something else to do with the pods.) Place them in the soup pot along with the diced potatoes, ginger, garlic, salt, and water, and half-and-half if you're using that (the soymilk is added later, if you're using that).

Bring to simmer over medium heat, cover, and cook about 10 minutes, or however long it takes for the potatoes and peas to become very tender. (The half-and-half - if you're using it - may curdle, but that's OK.) Transfer in batches to a food processor or blender, and puree until smooth. Add water to adjust consistency if desired. If using soymilk, add this now. Return to pot and re-warm for just another few minutes over low heat. Sprinkle with the mint and sesame seeds.