Melt butter in your pot over medium heat until it's completely melted and covering the whole bottom nicely.
Toss in onion, celery, and garlic, then cook for 5 minutes stirring often until everything gets soft and clear.
Add potatoes, broth, and bay leaf to the pot, bring it to a boil, then lower the heat right away.
Let it simmer for 15 minutes until you can easily poke a fork through the potatoes without any resistance.
Mix in ham and thyme, then keep simmering for 5 more minutes to warm the ham and blend all flavors.
Fish out the bay leaf, then use your stick blender to blend some of the soup, but leave some chunks.
Stir in cream on low heat, add salt and pepper to your taste, and mix it all up really well.
Heat the soup gently for 3 minutes, but don't let it boil or the cream might get weird and lumpy.
Scoop your soup into bowls, add some parsley on top if you want, and serve it hot with crusty bread.