This soup makes a delicious first course; serve it chilled or at room temperature.
Melt butter in a medium saucepan over medium heat. Add onion, leek, and celery; cook, stirring occasionally, until softened but not browned, about 7 minutes. Add 5 cups water; bring to a boil. Reduce heat, and simmer 10 minutes. Add peas, and return to a simmer; cook, stirring occasionally, until peas are tender but still bright green, 3 to 4 minutes. Remove pan from heat; stir in mint leaves.
Working in batches, puree vegetable mixture in a blender or food processor, filling no more than halfway each time. Season with salt and pepper.
Heat 1 tablespoon oil in a medium saute pan over medium-high heat. Add mushrooms; cook, stirring occasionally, until mushrooms are tender and liquid has evaporated, about 3 minutes. Transfer mushrooms to a plate; set aside.
Wipe out pan; add remaining tablespoon oil. Heat over medium heat. Add fiddleheads; cook, stirring occasionally, until bright green and slightly softened, 4 to 5 minutes. Transfer to a plate.
Ladle soup into serving bowls, dividing evenly. Garnish with mushrooms, fiddleheads, and small mint leaves.