Clean shiitake caps with soft paintbrush or damp towel. Slice stems thin and combine in a small pot with stock, thyme, and allspice, and 1 teaspoon fat or oil. Simmer, partly covered, until tender— about 10 minutes. Cut caps into generous mouthfuls, or leave whole, if small.
Heat remaining oil in a large skillet; stir in garlic and toss a moment. Add shiitake c