Place the boiling water in a mixing bowl and add the dried morels. Let them soak for about 30 minutes.
Transfer the mushrooms to paper towels using a slotted spoon.
Pour the mushroom soaking liquid through a sieve lined with a coffee filter or a paper towel into a saucepan. Simmer until reduced to about 1/3 cup. This should take about ten minutes.
Add a third of the morels and reserve. Finely chop the remaining mushrooms.
Preheat the oven to 375 F.
Use a large skillet to heat 1 1/2 tablespoons of oil over medium high heat. Sauté the shallots and garlic until they are soft. Transfer the mixture to a bowl and stir in the chopped mushrooms, bread crumbs, parsley, and salt and pepper to taste. Allow the stuffing to cool to room temperature before proceeding.
To make a hole lengthwise through the center of the pork loin, use a long, thin, sharp knife. Beginning at one end, work the knife into the loin and head toward the center of the loin. When you reach the middle, repeat the procedure starting at the other end of the loin. The hole should run lengthwise all the way through the middle of the loin. Using a wooden spoon, open up the incision to create a 1 1/2 inch opening. Working from both ends of the loin, pack the stuffing into the opening, pushing toward the center. Season the outside of the loin with salt and pepper to taste.
Use a skillet to heat the remaining 1/2 tablespoon of oil over high heat until it's just about hot enough to smoke. Brown the loin on all sides.
Transfer to a flameproof roasting pan and roast in the middle of the oven for approximately one hour. A meat thermometer inserted into the center of the meat should register 160 F. Transfer the pork to a cutting board and permit it to "rest" for about ten minutes.
While the loin is standing, add the lime juice to the roasting pan. Deglaze over moderate heat, scraping up brown bits. Add the veal demiglace, the reserved morels, and the water used for soaking the morels. Simmer, stirring, for about five minutes.
Serve hot with the pork loin.