Butterfly, and pound pork loin to even 1-inch thickness with mallet or bottom of heavy skillet.
For the brine: In a large, wide bowl, dissolve sugar and salt in 3 cups hot water. Add bay, allspice, peppercorns, garlic, and 5 cups cold water; stir to combine. Add butterflied and pounded pork; cover bowl with plastic wrap and refrigerate until fully seasoned, about
1 1/2 hours. Remove pork from brine, pick spices off meat, and dry pork thoroughly with paper towels.
For the stuffing and glaze: Once the pork is in the brine, adjust oven rack to lower-middle position and heat oven to 325 degrees. Process half the bread pieces in work bowl of food processor fitted with steel blade until broken into crumbs with few pieces no larger than about 1/4 inch, about 45 seconds; transfer to large mixing bowl and set aside. Repeat process with remaining bread pieces (you should have about 4 cups crumbs total).
In now-empty work bowl, process apricots, garlic, cumin, coriander, cinnamon, and cayenne until finely ground, about 30 seconds; add mixture to reserved bread crumbs. Add onion, figs, pistachios, thyme, parsley, salt, and pepper to taste to bread crumb and apricot mixture; toss until well distributed, breaking up any apricot clumps as necessary. Beat eggs and cream in small bowl; pour over bread and apricot mixture and toss with hands until evenly moistened and a portion of mixture holds together when pressed.
On parchment paper–lined cookie sheet or inverted rimmed baking sheet, form stuffing into log shape equal in length to butterflied pork. Cover stuffing with foil and bake until firm and cooked through and butterflied pork has been removed from brine and prepared for stuffing, about 45 minutes. Remove stuffing from oven; increase oven temperature to 450 degrees.
While stuffing bakes, heat apricot preserves in small saucepan over medium-low heat, stirring occasionally, until melted but not liquefied, 5 to 7 minutes. Strain through small strainer into small bowl (you should have about 1/3 cuand set aside; discard solids in strainer.
To stuff, roast, and glaze the roast: Line -shallow roasting pan or rimmed baking sheet with foil, position flat wire roasting rack over foil, and set aside. Stuff, roll, fasten, and tie pork loin. Place stuffed roast on rack, brush one-half apricot glaze evenly over exposed surface of meat and roast 20 minutes. Remove roast from oven and, with tongs, rotate roast so that bottom side faces up. Brush exposed surface with remaining apricot glaze; return roast to oven and roast 25 minutes longer (glaze should be medium golden brown and internal temperature of both meat and -stuffing should register 145 to 150 degrees on instant-read thermometer). Transfer roast to carving board, tent with foil, and let rest 5 minutes. Cut off twine, slice, and serve.