Combine all of the stock ingredients in a large pot. Bring to a boil, reduce the heat to medium, and simmer, partly covered, 20-30 minutes. Strain and reserve the stock.
Place the sliced leeks in a large bowl and fill with cool tap water. Swish the leeks occasionally to remove any sand from between their layers.
Melt the butter in a large saucepan over medium heat. Lift the leeks out of the water, give them a shake, and add them to the pot along with the diced potato. Saute, stirring frequently, until the leeks are tender, about 10 minutes.
Add 6 cups of the stock and the salt, bring to a simmer, and cook until the potatoes are tender, about 10 minutes.
Add the peas, and simmer for 5 minutes. Add the lettuce, and bring to a simmer for 1 minute. Remove from heat and let cool to room temperature.
Add the mint leaves, and puree the soup with a hand blender or in batches in a blender. Add the lemon juice. Taste, adjusting the salt or lemon if needed.
To serve, heat the soup gently, stirring frequently, over a medium flame until just warm. Ladle into bowls, and swirl with crème fraîche, peas, a few mint leaves, and a handful of croutons.
