In a heatproof bowl, add the dried porcini or shiitake mushrooms to the 3 cups boiling water. Give it a quick stir and let them soak until ready to use. You'll be using the soaking liquid as an ingredient in the dish.

Melt 2 tablespoons of the unsalted butter in a large high-sided ovenproof skillet, braiser, or short wide pot (at least 3½ qt.) over high heat. Add half of the mixed fresh mushrooms and cook, stirring occasionally, until the mushrooms release their liquid. The mixture should come to a simmer in about 5 minutes. Continue to cook, stirring occasionally, until liquid evaporates and mushrooms are golden brown, 6 to 8 minutes more. Using a slotted spoon, transfer to a medium bowl.

Repeat the process with another 2 tablespoons unsalted butter and remaining mixed fresh mushrooms.

Return the first batch of mushrooms to the skillet. Sprinkle the flour evenly over the caramelized mushrooms. Cook, stirring often, until well incorporated for about 2 minutes. The flour may clump but will smooth out later in the braising process. Transfer the mushrooms back to the medium bowl, scraping the pan to include as much flour as possible. Set aside until needed.

In the same skillet, melt 2 tablespoons of the unsalted butter over medium heat. Add the sliced carrots and finely chopped garlic along with the frozen pearl onions. Cook, stirring often, until the vegetables are softened around the edges, for about 6 to 8 minutes. Add the tomato paste and cook, stirring constantly, until deepened in color, about 3 minutes. Pour in the dry red wine, scraping up any brown bits. Cook, stirring occasionally, for about 5 minutes until reduced by about one-third.

Add the sprigs of fresh thyme, bay leaf, soy sauce and salt to the mixture. Hold a fine-mesh sieve over the skillet and pour the reserved dried mushroom soaking liquid through. Discard the the mushrooms that are remaining in the sieve. Season generously with freshly ground pepper.

Bring to a boil and cover the skillet. Reduce the heat so the mixture maintains a gentle simmer. Cook, stirring occasionally, until the carrots and onions are tender for about 45 to 50 minutes.

Uncover and return the reserved caramelized mushrooms to the skillet. Cook, uncovered over low-medium heat, stirring occasionally, until mixture thickens slightly and mushrooms are tender, for 18 to 20 minutes.

Meanwhile, heat the olive oil in a large nonstick skillet over medium-high heat. Add 8 to 12-ounces of the maitake or oyster mushrooms in a single layer. Cook, undisturbed, until deeply browned and crisp underneath, about 5 minutes. Turn and cook until other side is golden, about 4 minutes. Set aside to be used just before serving.

Stir the remaining 2 tablespoons unsalted butter into the mushroom bourguignon. If using, add the desired amount of chopped fresh parsley. Remove from heat and if needed, season with more salt and/or pepper. Remove the thyme and bay leaf; discard.

Spoon the crispy maitake or oyster mushrooms on top. Alternatively, the crispy mushrooms can be reserved to top off each individual serving.

To serve, divide buttered cooked egg noodles, mashed potatoes or cooked polenta among plates for individual servings. Spoon mushroom bourguignon over; drizzle each serving with a little sherry wine vinegar or red wine vinegar. If desired, sprinkle some finely chopped parsley over the top.

Do Ahead: Mushroom bourguignon (without the crispy mushrooms) can be made a day ahead. Transfer to an airtight container; chill. Bring to a simmer over medium, adding splashes of water or broth to loosen as needed, before serving.