Step 1Place the butter and the prosciutto in the bottom of a large saucepan over medium-low heat. Cook until the prosciutto softens and begins to darken, 3 to 4 minutes. Add the leek and shallot, and cook until they are soft and glistening, about 5 minutes. Add the white wine, and reduce to a syrup. (Recipe can be made to this point several hours in advance.)
Step 2Add the half-and-half, and bring just to a simmer over medium-low heat. When you see bubbles beginning to appear around the rim of the pan, add the oysters. Cook just until the oysters are plump and firm, about 3 minutes. Season to taste, if desired. Remove from the heat and ladle into well-heated soup bowls. Garnish each serving with one-fourth teaspoon chopped tarragon and serve immediately.