How to Make It
In a small heatproof bowl, cover the dried porcini with 1 cup of boiling water and let stand until softened, about 15 minutes. Drain the porcini and reserve the soaking liquid. Finely chop the porcini.
In a large skillet, melt 2 tablespoons of the butter. Add the porcini and shiitake and season with salt and pepper. Cook over high heat, stirring occasionally, until the shiitake are softened and lightly browned, about 8 minutes. Add the shallot and garlic and cook until fragrant, about 2 minutes. Add the porcini soaking liquid, stopping before you reach the grit at the bottom. Boil until the liquid is evaporated, about 5 minutes. Scrape the mixture into a bowl.
Add the remaining 2 tablespoons of butter to the skillet. Add the cabbage, season with salt and pepper and cook over moderately low heat, stirring frequently, until the cabbage is very tender, about 15 minutes. Add a few tablespoons of water to the skillet if it looks dry. Return the mushroom mixture to the skillet. Add the cream and tarragon and simmer until the cream is nearly evaporated, about 5 minutes.
Heat a cast-iron grill pan. Brush the bread with oil and grill over high heat, turning once, until toasted. Spread the cabbage-mushroom mixture on the toasts and serve.