1. In a large glass measuring cup, soak the porcini in the boiling water until softened, about 20 minutes. Drain the porcini, reserving the soaking liquid. Rinse them briefly, pat dry and finely chop. Let the porcini soaking liquid settle, then pour it into a medium bowl, leaving the grit behind.
2. Preheat the oven to 350°. In a medium skillet, melt 2 tablespoons of the butter. Add the onion and cook over moderately high heat, stirring, until translucent, about 5 minutes. Add the garlic, chopped porcini and 2 tablespoons of the reserved porcini soaking liquid and cook over moderately low heat for about 5 minutes. Stir in the rosemary, thyme and sage and scrape the mixture into a large bowl; let cool.
3. Soak the bread in the remaining porcini liquid until evenly moistened, then squeeze dry and tear into small pieces. Reserve the porcini liquid for the gravy. Add the soaked bread to the large bowl, along with the ground beef, veal and pork and the ham and eggs. Season with 1 tablespoon of kosher salt and 1 teaspoon of black pepper. Knead the mixture with your hands until combined. Transfer the meat to a large enameled cast-iron baking dish or a small flameproof roasting pan and pat it firmly into a 10-by-6-inch oval loaf.
4. In a bowl, mix the mayonnaise with the mustard and cayenne and brush it over the meat loaf. Bake for 1 1/2 hours, or until browned and firm to the touch. Using 2 large spatulas, transfer to a cutting board, cover loosely with foil and keep warm. Set aside the baking dish.
5. Meanwhile, melt the remaining 1 tablespoon of butter in a medium skillet. Add the shiitake mushrooms and cook over moderate heat, stirring frequently, until softened, about 5 minutes. Add the shallot and cook until translucent, about 3 minutes. Add the chicken stock and porcini soaking liquid and simmer over moderate heat until reduced by half, 8 to 10 minutes.
6. Place the meat-loaf baking dish over moderately high heat and stir occasionally until the drippings start to brown, about 3 minutes. Add the wine and cook, scraping up any browned bits, until reduced by half, about 3 to 4 minutes. Strain the contents of the baking dish into the skillet, whisk in the flour mixture and boil until thickened, about 2 minutes. Season the gravy with salt and pepper. Cut the meat loaf into thick slices and pass the mushroom gravy at the table.
Make Ahead: The meat loaf and gravy can be kept refrigerated separately for up to 2 days.
Wine Recommendation: An intense, peppery and full-bodied Petite Syrah would pair extremely well with this earthy, herb-filled meat loaf. Look for the 1997 Foppiano Petite Syrah from California or the 1996 Stags' Leap Winery Petite Syrah.