For the brine: To a large pot, add the water and bring to a boil. Once at a boil, add the garlic, agave syrup, red chile flakes, peppercorns, and salt and turn off the heat. Once slightly cooled, add the mixture to a large container and add 6 cups ice to yield about 10 cups liquid. Add the pork and submerge. Place into the refrigerator for about 2 to 3 hours. Remove and rinse under cool water; pat dry with paper towels.
Preheat the oven to 350 degrees F.
On a steady cutting board (place a couple of damp paper towels under it to keep it from moving around), stand the roast up with the bones up and the loin down. With a sharp, thin knife, slice carefully along the bones, separating the eye of the loin from the bones, but leaving the loin still attached on the bottom. Continue butterflying and slowly roll out, creating a large flat surface for the pesto to easily spread on.
For the pesto filling: In a small bowl, combine the pesto, chile flakes, garlic, salt and pepper. Mix well and spread this mixture evenly on the inside of the rolled out loin. Carefully roll back up. Cut a 2 to 3-inch slit in between the bones. Truss the roast with kitchen string in between the bones and around the roast. This will help the roast cook evenly.
For the roasting vegetables base: In a heavy, large roasting pan, add the onions, carrots, parsnips, Brussels sprouts, and smashed garlic and toss to combine. Place the tied pork roast on a v-rack over the vegetables, tent with heavy duty aluminum foil, and roast in the oven until the internal temperature reads 145 to 150 degrees F, 2 1/2 to 3 hours. About 3/4 of the way through the cooking process remove the foil, this will help the pork brown up a bit. Remove from the oven, tent with foil, and let sit for 15 to 20 minutes to rest and carry over cooking for another 5 to 10 degrees F. Place the roasting pan over medium-high heat and add 1 1/2 to 2 cups chicken stock and bring to a boil. Allow to boil and reduce for 5 minutes.
To finish: Strain the vegetables and place onto a serving platter. Finish the sauce with about 2 tablespoons butter and remove from the heat.
Remove the kitchen string and slice the pork into 1 1/2-inch thick pieces. Serve immediately with the vegetables and pan sauce.