Step 1Wipe the mushrooms clean, trim any hard stems and cut them into roughly almond-size pieces. Try to use a mixture of sizes of mushrooms so you get a diversity of shapes. The small ones can be left whole, those that are a little bigger can be cut in half, and so on.
Step 2Heat 2 tablespoons of the butter in a skillet over medium-high heat until the foam has subsided and the butter turns a light hazelnut color. Add the mushrooms, sprinkle with half a teaspoon of salt, cover tightly and cook, tossing occasionally until the mushrooms begin to glisten and give up their moisture, about 3 minutes. Remove the cover, add the garlic and the parsley, raise the heat to high, and continue cooking, stirring constantly until the mushrooms are richly aromatic and soft, but not flaccid, about 3 minutes.
Step 3Transfer the mushrooms to a bowl. Add the white wine to the skillet. Cook over high heat until it reduces to a syrup, about 2 minutes. Strip the leaves from the thyme between your thumbnail and finger and add them to the syrup along with the cream. Cook, stirring to incorporate into a smooth liquid. Add the mushrooms back to the sauce, toss to coat well and set aside.
Step 4Cut the potatoes into half-inch pieces and steam in a tightly covered pot over rapidly boiling water until they are just tender, about 15 minutes.
Step 5Warm the mushroom mixture over medium heat and add the potatoes as soon as they are done. Do not let the potatoes cool or they won't absorb the flavors. Add the sherry vinegar and stir everything together. Adjust the salt seasoning (it will definitely need more after you've added the potatoes) and season with a grinding of fresh black pepper. (The dish can be prepared to this point up to 1 hour in advance.)
Step 6Before serving, warm the mixture through over medium-high heat, add the remaining 1 tablespoon of butter and stir to mix well.