Heat 3 tbsp. butter in a 3-qt. high-sided skillet over medium heat. Add button mushrooms and mushroom trimmings and cook, stirring often, until they release their liquid, about 10 minutes. Add 3⁄4 oz. dried shiitakes and 4 cups of water; boil. Reduce heat to medium-low and simmer until liquid has reduced to 3 cups, about 10 minutes. Set a fine-mesh strainer over a 2-qt. saucepan. Strain broth, pressing mushrooms with the back of a spoon to extract liquid; discard solids. Add cream to mushroom broth and bring to a boil. Reduce heat to medium-low and simmer until flavors meld, about 10 minutes. Season broth with salt and a pinch of cayenne and set aside.
In a spice grinder, grind remaining dried shiitake mushrooms to a fine powder. Transfer mushroom powder to a small skillet over medium-high heat and toast, swirling pan constantly, until fragrant, about 5 minutes. Transfer mushroom powder to a small bowl; set aside.
Heat remaining butter in the skillet over medium heat. Add mixed mushrooms, season with salt, and cook, stirring gently, until tender, 4-5 minutes. Add crayfish, season with salt, and cook until hot. Remove pan from heat and set aside.
To serve, foam reserved broth on high speed in a blender or with the steamer attachment on an espresso machine. Mound crayfish mixture in 4 teacups or small bowls, and ladle in broth. Spoon foam on top, dust with mushroom powder, and garnish with chervil.