Melt butter in a large saucepan over medium heat. Add onion, leek, and celery, and cook until soft. Add potatoes and 7 cups vegetable stock; bring to a boil. Reduce heat and simmer until potatoes are soft, 10 to 15 minutes. Remove from heat.
Prepare an ice-water bath. Fill another large saucepan with water and bring to a boil over high heat. Add salt and return to a boil. Add peas and cook until tender, about 5 minutes. Drain and immediately transfer peas to ice-water bath to cool. Drain and add to saucepan with vegetables and stock.
Working in batches, puree the soup in the jar of a blender or bowl of a food processor. Strain liquid twice through a fine mesh sieve into another large saucepan. Add 1 cup of heavy cream and place saucepan over low heat. Season with salt, pepper, and sugar. Tie mint together using kitchen twine and add to saucepan; let steep for 10 to 15 minutes. Remove mint and discard. If soup seems too thick, add some of the remaining vegetable stock to thin.
In a medium bowl, whisk remaining 1/2 cup heavy cream until soft peaks form. Gently fold into the warmed soup. Serve immediately.