In a deep rectangular or oval dish that can contain the meat and all the other ingredients except for the mushrooms, combine the pork tenderloin, 3 tablespoons of olive oil, onion, garlic, bay leaves, celery, rosemary, and red wine. Turn the meat over several times to coat it well, then cover the dish with plastic wrap and refrigerate overnight. Take it out occasionally whenever convenient to turn the pork over, basting it with its marinade.
The following day, take the meat out of the refrigerator at least 1 hour before proceeding with the preparation of the dish. Turn it over and baste it when you take it out of the refrigerator, and once every half hour thereafter.
Detach the mushroom caps from the stems, discarding the stems. Wash the caps quickly in running cold water without letting them soak. Pat them dry gently but thoroughly with a cloth towel, and cut them into thin slices.
Lift the tenderloin out of the deep dish, pick out any bits of the vegetables from the marinade that may be sticking to it, and pat the meat dry with kitchen towels.
In a skillet that can accommodate the two pieces of pork without overlapping, put in the remaining tablespoon of olive oil, and turn the heat to high. When the oil is hot enough to sizzle when you put in the meat, slip in both pieces. Turn the meat over to brown it evenly all around, then transfer it to a platter.
Pour all the marinade from the deep dish into the skillet, turn the heat down to low, and cover the pan. Cook, stirring occasionally, until all the vegetables are very soft or almost dissolved.
While the marinade is cooking, put 1 tablespoon of butter in a medium skillet, turn the heat to medium, and add the sliced shiitake caps with some salt. Cook, turning the mushrooms over occasionally, until the liquid they shed evaporates completely and they have become very tender.
When the vegetables of the marinade are very soft, add the cooked shiitake mushrooms, cooking them together for about a minute or two. Add both pieces of pork, sprinkling them with salt and several grindings of black pepper, and raise the heat to high. Cook the meat for 10 minutes on each side.
Transfer the meat onto a cutting board, cut into slices 1/3-inch thick, and place slices on a very warm serving platter. Remove the bay leaves from the marinade -- and the garlic cloves, if you can find them -- then cover the meat with the cooked marinade and mushrooms and serve at once.
Ahead-of-time note: When the overnight marinating of the meat is complete, it would be desirable to proceed at once through all the succeeding steps, but if you wish you can stop a few hours in advance right after the shiitake mushrooms and cooked-down marinade have been combined. Cook the meat, however, only when ready to serve, because it may dry out and become stringy if reheated.