Place a heavy bottomed pot or rondeaux over medium-high heat. Add the butter and olive oil, followed by the sliced shallots and elephant garlic. Reduce the heat to medium and continue to sauté until the shallots have just begun to soften and color. Add the hawaij, porcini powder and flour, then stir to coat. Continue stirring, minding the bottom, and cook for two minutes. Add the mushrooms and cook until they have released most of their water (about 6-8 minutes). Add the chardonnay, sherry and lemon juice. Cook for three minutes. While stirring, add the stock, then reduce the heat to a bare simmer and continue to cook for 20 minutes.
Allow the soup to cool enough to safely blend (about 15 minutes) or chill in an ice bath. Using an immersion or standard blender, puree the soup base smooth. You may also pass the soup through a medium strainer, if desired. The soup base may be held under refrigeration for up to three days tightly covered.
When ready to serve, bring the soup to a simmer over medium heat. Whisk in the creams and cook for a final 10 minutes. Taste and adjust for seasoning and acidity (add more lemon juice if necessary). Divide between eight bowls and garnish with lightly pickled mushrooms, porcini oil, lemon zest and fresh tarragon.