Trim leek roots and trim tops leaving about 2 ½-inches of green. Cut a cross through root end and thoroughly wash removing any sand/dirt. Slice leeks finely.
Heat oil in a Dutch oven or stockpot over medium heat. Add leeks and onion, stirring occasionally, until they are softened and taking on a golden color.
Add potatoes, broth, salt and pepper. Cover and simmer for 40 minutes.
Puree in a blender or food processor; return to pot.
Whisk flour or cornstarch with ½ cup of milk.
Add to pot with remaining milk, stirring constantly for 5 minutes.
Taste and correct seasoning with salt and pepper.
Strain soup through a fine sieve to insure a super smooth texture, if desired.
Soup can be served warm or chilled topped with garnishes.
Notes
Serves 6 or many more as shooters. To easily fill shot glasses pour soup from a measuring cup or other vessel with a spout.