Heat extra-virgin olive oil in a large pot over medium-high heat. Add in your leeks, tossing to sweat and soften, about 2 to 3 minutes.
Next, add your garlic, salt, and pepper, tossing until fragrant, another 2 to 3 minutes.
Stir in your potatoes, season with salt and pepper to taste, and add your broth.
Bring the heat to medium, medium-low and bring this mixture to a simmer until the potatoes are fork-tender.
Transfer to a blender, or use an immersion blender until completely smooth. If using a blender, be sure to wait a couple minutes until the mixture is cool, and use the release setting to allow air to vent through.
Transfer back to the pot and with the heat off, mix in your heavy cream and vinegar.
Serve in soup bowls, topping with your chopped chives, a drizzle of olive oil, cream, and freshly cracked black pepper.
Notes
Wash your Leeks: Ensure you are washing your leeks after slicing them in a bowl of water, since leeks are very dirty by nature! Dry well before tossing into the base of the soup.
Make This Vegan and Dairy-Free: Swap the heavy cream with coconut milk (or leave it out completely).
Vinegars: We are using sherry vinegar to finish off the soup, but you can substitute it with red wine vinegar or white wine vinegar.
Storing: Store in an airtight container in the refrigerator for up to 4 days.