Rub each loin roast with a portion of salt. Place on a wire rack set over a pa­per-towel-lined jelly roll pan and refrigerate, uncovered, for 5 days. Before roasting meat, shave off very thin exterior layer of hardened, dehydrated meat on each end of each loin.

Preheat oven to 250ºF with a rack in the lower-middle of the oven. Mix garlic, pepper, thyme and 2 tablespoons olive oil to make a thick paste. Rub each loin with a portion of paste.

Toss carrot and onion with 1 tablespoon olive oil and spread in the bottom of a roasting pan. Place a baking rack over vegetables and arrange loins on rack. Roast until inner temperature of loins is 120° to 125°F, about 1 ½ hours. Increase oven temperature to 425°F and continue to roast until meat registers 145°F, about 30 minutes more, adding 1 cup water to roasting pan once pan drippings turn brown and just start to smoke. (Internal temperature will continue to rise after loins are removed from oven.)

Transfer each loin to a serving platter. Cover loosely with foil while making sauce. Strain pan drippings into a measuring cup, pressing on vegetables to release liquid, and skim fat. Add additional water, if necessary, to equal 1 cup. Transfer to a small saucepan, add Madeira and simmer to blend fla­vors, about 5 minutes. Carve roast, cutting between each rib, and pass jus separately at table.