Place the roast onto a cutting board with the short side toward you and fat facing down. Using a sharp filet knife (with the blade horizontal to the board), butterfly the roast open about 1 inch from the bottom of the roast. Work carefully to not cut all the way through. Continue to slice horizontally, opening the roast up until you have a large, thin, flat piece of meat. Place the pork in large airtight freezer bag (or large plastic container with a lid).
To prepare the brine, bring 6 cups water, the salt, sugar, peppercorns, garlic, thyme, bay leaves and onions to a simmer in a medium pot over medium heat. Stir until the sugar and salt have completely dissolved. Remove from the heat and cool by adding the ice cubes. Pour the brine into the freezer bag, submerging the roast completely, seal and set aside for 25 minutes.
Preheat the oven to 375 degrees F.
Remove the pork from the brine and pat dry with paper towels. Lay the sausages inside lengthwise. Pierce the casings with the tip of a sharp paring knife (this will ensure the flavor of the sausage goes into the pork as it cooks). Starting with the short side, roll up the roast tightly, jelly roll style. Secure the roast with tightly fitted kitchen twine at even intervals about 2 inches apart. Sprinkle with salt, pepper, the granulated garlic and paprika.
Heat the oil in a large cast-iron pan over medium heat. Add the roast and sear fat-side down first, 3 to 4 minutes, and then on each of the three other sides. When turned to the final side, transfer the pan to the oven and bake until the internal temperature hits 145 degrees F on an instant-read thermometer, 45 to 55 minutes. Remove the roast from the oven; rest the pork on a platter while you make gravy from the pan drippings.
Set the pan over medium-low heat. Add a little oil, if needed, and saute the onions, carrots and celery. Cook the vegetables until well caramelized, about 8 minutes. Then season with salt and pepper. Deglaze the pan with the vermouth and reduce until the liquid has almost completely reduced. Add the chicken stock and reduce again for another 5 minutes; this will ensure you have a nice, rich sauce. Taste and season with salt and pepper.
Remove the pan from the heat and swirl in the butter to thicken the sauce. Finish with the lemon zest, a squeeze of lemon juice and serve the sauce over sliced andouille-stuffed pork loin.