Add potatoes, onions, celery, salt, pepper, savory, and 8 cups water to a large pot on high heat. Boil until potatoes are tender, about 20 minutes. Remove from heat and let cool slightly.

Stir in heavy cream and buttermilk. Return the pot to stove on medium-low heat.

Make a cornstarch slurry by whisking together cornstarch and 3 tbsp water. Add to the soup and whisk to combine. Heat while whisking until desired thickness is reached. Garnish with chopped parsley if desired. Serve hot.