Pour olive oil into a large pot, set over medium heat, and add leeks to hot oil. Season leeks with salt. Cook, stirring often, until leeks are soft (about 5 min). Pour chicken broth on leeks; bring to a boil. Season with black pepper and cayenne pepper.
Stir peas into boiling broth mixture and add 2 mint leaves. Bring back to a boil and cook until peas are bright green and have started softening (about 5 min).
Turn off heat; use a slotted spoon to transfer vegetables into a large blender. Leave liquid in the pot. Pour half liquid from the pot into blender, place a folded towel on blender lid, and pulse a few times to chop vegetables. Blend at high speed to puree vegetables. Add remaining liquid from pot to blender and blend soup until smooth.
Pour soup from blender into a fine mesh strainer set over a bowl and use a spatula to push soup through strainer. Remove fibers left behind by veggies. Cool soup to room temperature, cover with plastic wrap, and refrigerate until chilled (about 3 hours).
Season cold soup with salt and cayenne pepper. Ladle soup into a serving bowl. Squeeze desired amount of sour cream on top. Serve garnished with fresh mint.