Preheat oven to 400°. Melt butter in a large pot over medium heat. Add leek and cook, stirring, until soft and translucent, about 6 minutes. Add garlic and cook 2 minutes longer. Put sweet potatoes in pot; add 3 cups water, 1 1/2 tsp. salt, and 1/2 tsp. pepper. Increase heat to high, bring to a boil, then reduce heat to medium-high and simmer until sweet potatoes are tender, about 10 minutes. Remove from heat and let cool 10 minutes. Working in batches, purée soup until smooth.
Prepare the garnish: Spread prosciutto slices on a rimmed baking sheet in a single layer. Bake until crisp, 7 to 10 minutes (watch carefully, as they can burn quickly). Let prosciutto cool completely on baking sheet (about 1 hour), then crumble into tiny pieces and set aside.
Transfer puréed soup to a clean pot set over medium-low heat. Stir in cream and up to 2 cups water (enough to make soup easy to drink out of cups). Season to taste with salt and pepper. Serve soup hot, in demitasse cups, garnished with prosciutto or chives if you like.
Note: Nutritional analysis is per 1/3-cup serving with 1/2 tsp. prosciutto.