1. In a large stockpot set over medium heat, combine the vegetable broth, potatoes, onion, celery, bay leaf and thyme sprigs. Simmer until the vegetables are tender, about 20 minutes. Carefully remove the bay leaf and thyme sprigs and add the cucumbers. Simmer for 5 minutes more. Turn off the heat and use a hand blender to purée the soup until smooth. Stir in the cream and season to taste with salt and pepper. Allow the soup to cool completely, at least 45 minutes, or transfer to a covered container and refrigerate overnight.

2. In a medium saucepot set over high heat, heat the oil to 350°. Carefully add the parsley and fry for a few seconds; the parsley should still be bright green. Remove the parsley from the hot oil with a fine-mesh strainer and drain on paper towels.

3. Return the soup to the stockpot and heat over medium heat until almost boiling. Stir in the milk and remove from the heat. Use a hand blender to briefly whip the soup. Pour the soup into bowls, garnish with parsley and serve immediately.