Cut and Clean Leeks. Cut off the roots and the dark green parts of the leek and discard them. Slice the white and light green parts of the remaining leeks into circle slices. Add them to a large bowl, pushing out the little rings of each slice slightly as you add them to the bowl. Fill the bowl with water and the swish around the leeks a few times before letting them sit for a minute. Use your hands to remove the leeks and place into a colander because the dirt from the leeks will settle to the bottom of the bowl. Give a quick rinse.
Start the Soup. Add butter to a large pot on medium and add celery and onion. Saute for just 2-3 minutes until onions start to turn transluscent, but you don't want them to brown or caramelize.
Add Garlic, Leeks, and Spices. Add the garlic and saute another minute before adding leeks, celery salt, kosher salt, white pepper, and ground mustard and sauteeing for another 2 minutes.
Add Potatoes with Stock. Add the final chopped potatoes to the pot then add the stock of choice. Add thyme. Turn up heat to bring it all to a boil, and then turn down to simmer.
Let it Simmer. Let soup simmer for 15 minutes or until the potatoes are fork tender.
Remove Thyme. If using fresh thyme, discard the bunch of sprigs.
Blend. Use an immersion blender, or in batches, use a high powered blender to blend until all of the potatoes, leeks, onions, etc. are creamy.
Ladle, Garnish, and Serve. Add a tablespoon of half and half or heavy cream to each bowl, and top with crunchy croutons and/or green onions.