Preheat the broiler and position a rack 8 inches from the heat. In a large skillet, melt 1 tablespoon of the butter. Add 1 garlic clove and half of the chopped shallot and cook over moderately high heat until fragrant, about 1 minute. Add the spinach and cook, tossing, until wilted. Add the cream and cook over moderately high heat until thickened, about 7 minutes. Add the cayenne and season the spinach with salt and pepper; keep warm.
Meanwhile, in a medium skillet, melt 1 tablespoon of the butter. Add the remaining garlic clove and minced shallot and cook over moderate heat until fragrant, about 2 minutes. Add the shiitake, season with salt and pepper and cook, stirring occasionally, until tender, about 4 minutes. Remove from the heat. Stir in the remaining 2 tablespoons of butter and the bread crumbs.
Lightly brush a large baking sheet with oil. Set the flounder fillets on the sheet and pat the mushroom mixture onto each fillet. Broil the fish for 4 to 5 minutes, or until cooked through. Transfer the fish to plates and spoon the creamed spinach alongside. Sprinkle with parsley and serve.