In a large pot, warm the 2 tablespoons olive oil over medium heat. Add the 2 medium leeks and season with Salt and pepper. Cook, stirring occasionally, until softened, about 4 to 5 minutes. Add the 2 garlic cloves and stir until fragrant, 1 minute.
Add 4 cups vegetable broth and 1 ½ pounds potatoes, then bring to a boil over medium-high heat. Cover and turn the heat down to medium-low, then simmer until the potatoes are completely tender, about 15 minutes.
Using an immersion blender or by carefully transferring to a blender in several batches, puree the soup until smooth. Stir in the 1 cup milk, taste for salt and pepper, and serve hot.
Notes
What kind of potatoes are best? You can use any kind of potatoes for this soup! I used russet potatoes in these photos - they are a nice economical option - and I also like using Yukon gold potatoes for an ultra-creamy soup.
Cook the potatoes thoroughly. Similar to when making mashed potatoes, you want these potatoes to cook completely! I let them go until they are almost falling apart (no problem, because you will puree the soup) so that there aren't any lumps in the final soup.
Can I make this vegan? Absolutely. Substitute your favorite non-dairy milk for the milk to make vegan potato leek soup.
What about the cream? I like things on the lighter side, so I prefer milk instead of cream. However, cream is a traditional choice, and it makes the soup nice and rich. You can use cream or half-and-half if you prefer.
Add veggies. Feel free to branch out and add spinach to the soup pot with the potatoes before pureeing, or sautee an onion along with the leeks and garlic.
Use water. If you don't have broth on hand, just use water! The soup will still be plenty flavorful.