In a small bowl soak morels in boiling water for 30 minutes and transfer with a slotted spoon to paper towels to drain. Pour soaking liquid through a sieve lined with a dampened coffee filter or paper towel into a small saucepan and simmer until reduced to about 1/3 cup, about 10 minutes. Add one third morels and reserve. Finely chop remaining morels. Morels may be prepared up to this point 1 day ahead and chilled, covered.
Preheat oven to 375°F.
In a large skillet heat 1 1/2 tablespoons oil over moderately high heat until hot but not smoking and saute shallot and garlic until softened. Transfer mixture to a bowl and stir in chopped morels, bread crumbs, parsley, and salt and pepper to taste. Stuffing may be made 1 day ahead and chilled, covered. Bring stuffing to room temperature before proceeding.
To make a hole for stuffing that runs through center of pork loin, pat loin dry and, beginning in the middle of one end, with a long, thin, sharp knife make a lengthwise incision toward center of loin. Repeat procedure from opposite end of loin (to complete hole running through middle). With handle of wooden spoon or your fingers open up incision to create a 1 1/2-inch wide opening. Working from both ends of loin, pack stuffing into opening, pushing towards center. Season outside of loin with salt and pepper.
In skillet heat remaining 1/2 tablespoon oil over high heat until hot and just about smoking, and brown loin on all sides, about 1 1/2 minutes total. Transfer loin to a flameproof roasting pan and roast in middle of oven for about 1 hour, or until a meat thermometer inserted in center of meat (but to the side of the stuffing) registers 160°F. Transfer loin to a cutting board and let stand 10 minutes.
While loin is standing, on top of stove add lime juice to roasting pan and deglaze over moderate heat, scraping up brown bits. Add stock or demiglace and reserved morel liquid with morels and simmer sauce, stirring occasionally, 5 minutes.
Slice pork loin and serve with sauce.