Trim the loin so that there is only ¼ inch fat on top. Mix the salt, sugar, water, bay leaves, allspice, and thyme in a pan and heat until the salt and sugar are dissolved; let cool. Put the loin in a pan just large enough to hold it and the liquid and pour the brine over the pork. Let marinate overnight or at least 6 hours in the refrigerator.
When the pork is fully brined, remove it and wipe it dry. With the handle of a wooden spoon or with a round knife-sharpening steel, poke a hole carefully through the very center of the loin.
Put the garlic, shallots, 1 tablespoon of the oil, and 1 tablespoon water in a small sauté pan. Cover and sweat over low heat for 10 minutes; do not let the mixture brown. Stir in the ham and half the rosemary. Set aside to cool. When it is cool, push the mixture into the center of the loin so that it is evenly distributed. Mix the remaining oil and rosemary and rub all over the loin. Season and let marinate at room temperature for 2 hours.
Heat the oven to 325°F.
Heat a sauté pan over high heat and sear the loin and brown it on all sides, about 5 minutes. Put the loin in a roasting pan just large enough to hold it and cook for 30 minutes. Remove the loin and let sit for 20 minutes in a warm place, covered with a piece of foil.
Pour any fat out of the pan and wipe it gently to remove all the fat. Pour in the stock and reduce it to ½ cup, scraping the pan to dissolve any of the browned meat juices. Strain the stock into a saucepan. Whisk in the mustard and then the butter until it is incorporated. Keep the sauce warm.
Cut the pork into ⅛-inch-thick slices. Put the slices on hot plates and pour the sauce over them.