Remove dark green tops of leeks. Slice from top to bottom (basically split down the middle) and then rinse well to be sure you’ve removed all of the dirt. Then, coarse chop into little C’s.
Melt oil of choice over medium heat in a medium to large soup pot. Once it melts, add leeks and thyme. Saute for about 5 minutes. If the leeks start to burn while you’re cooking them, simply add a little broth to deglaze.
Add the sea salt, pepper and potatoes. Stir to combine for 1-2 minutes.
Add broth and bring to a boil.
Once boiling, reduce heat and simmer for 15-20 minutes, or until the potatoes are tender.
Remove from heat. If you have an immersion blender, blend soup until creamy and smooth (until you can’t see any whole potatoes in the soup). If you don’t have an immersion blender (consider getting one if you like smooth soups!) pour contents of the soup, likely in a few different batches, into your blender (see note). Serve immediately or keep in the fridge for up to 5 days.