Potatoes and leeks make for a versatile pairing of ingredients. In the three overlapping recipes below, we'll adapt a simple basic procedure to very different ends. 

The simple version of potato-leek soup is a great recipe if you're trying to learn to cook -- pretty fool-proof.

Then we'll add a couple of levels of complexity, with a chilled soup (Vichyssoise), perfect for summer, and finally, a more involved version (tho' still pretty easy), Roasted Garlic Potato-Leek Soup.

These recipes are inherently vegetarian; if you choose olive oil instead of butter when the option is offered, they're vegan. And if you want to add meat, the final version below suggests optional ham or bacon. 

Simple Potato-Leek Soup

Let's start with a dead simple Potato-Leek Soup, a peasant dish from the Country French canon.

3 tablespoons of butter or olive oil
1 pound leeks, with tough green parts removed 
1 pound potatoes, peeled and cut into 1/2 inch dice.
Salt
2 quarts water.

Clean the trimmed leeks. If you've never cleaned leeks, this video shows how. Once they're clean, slice the leeks thin.
 
Melt butter in your large soup pan over medium heat. Add leeks, stir to coat, reduce heat and cook until leeks are soft, 10-15 minutes, stirring often to prevent browning.

Add potatoes, salt to taste, and water. Bring to a fast simmer and cook until the potatoes are very tender, about 20-30 minutes. Taste for seasoning and add more salt if needed.

It's ready to eat as is, but you can also thicken the broth by mashing some of the potatoes with the back of a spoon then stirring into the soup.

Vichyssoise

That version is great on a cold winter day; this one hits the spot in the heat of a Louisiana summer.

3 tablespoons of butter or olive oil
1 pound leeks, with tough green parts removed, then cleaned as above 
1 pound potatoes, peeled and cut into 1/2 inch dice.
Salt
2 quarts chicken stock/broth
1/2 cup heavy cream
Fresh herbs for garnish (chives, tarragon, flat leaf parsley)

Prepare the recipe as above, except use 2 quarts of chicken broth/stock instead of water. Allow to cool enough to handle easily. Then puree the potato/leek/broth until smooth, working in batches if necessary. 

Once all the soup has been pureed and returned to a clean pan, chill in refrigerator. 

Before serving, stir in 1/2 cup heavy cream, and sprinkle with freshly chopped chives or parsley or tarragon.

Roasted Garlic Potato Leek Soup

Finally, another hot version, Roasted Garlic Potato leek soup

2 tablespoons butter 
2 tablespoons olive oil
1 pound leeks, with tough green parts removed, then cleaned as above
2 medium onions, yellow or white, peeled and cut into medium dice.
2 heads garlic
1 pound potatoes, peeled and cut into 1/2 inch dice.
Salt
2 quarts chicken stock/broth
1/2 cup heavy cream
Fresh herbs for garnish (chives, tarragon, flat leaf parsley)

Roast two heads of garlic and set aside to cool. When cool enough to handle, squeeze the soft garlic into a small bowl, add a drizzle of olive oil, and stir into a smooth puree with a fork. If you've never roasted garlic, see how in this video. 

Meanwhile, heat olive oil and melt butter in large soup pan over medium heat. Add leeks and chopped onions, stir to coat, reduce heat and cook until the vegetables are soft, 10-15 minutes, stirring often to prevent browning.

Stir in the roasted garlic puree. Then add potatoes, salt and chicken stock. Bring to a fast simmer and cook until the potatoes are very tender, about 20-30 minutes. Taste for seasoning and add more salt if needed.

Take the soup off heat until it's cool enough to handle, the puree, working in batches if necessary. Return the soup to the pan over low heat.

Optionally, add crumbled cooked bacon or finely chopped ham. 

Add chopped herbs as a garnish.