Tie the thyme sprigs, parsley stems, and bay leaf into a bundle with cooking twine.
Melt the butter in a large pot over medium heat. Add the celery, onion, and leeks to the pot and season with 1/4 tsp salt and 1/8 tsp pepper. Cook, stirring occasionally until onions are translucent.
Add the wine and simmer, about 5-6 minutes.
Add the stock, potatoes and herb bundle and bring to a boil over high heat. Reduce heat to medium-low, and simmer partially covered (about ¾ of the opening of the pot should be covered) about 45-50 mins, or until potatoes are very tender.(Reduce heat if needed to maintain a low simmer.)
Remove the pot from the heat, and remove the herb bundle. Use an immersion blender to puree the soup, or, allow the soup to cool, then puree in a blender.
Stir in milk to taste. Taste the soup for seasoning and add more salt and pepper if desired. Top the soup with the chopped parsley leaves, thyme leaves, and crispy leeks.