In a large saucepan, combine the cider, cinnamon, peppercorns, coriander, crushed red pepper, garlic and 3 tablespoons of kosher salt and bring to a boil, stirring to dissolve the salt. Remove from the heat, add the ice and let cool completely. Pour the brine into a bowl and add the pork tenderloins, then cover and refrigerate overnight.

Preheat the oven to 450°. In a small saucepan, combine the vinegar with the shallot, thyme and sorghum syrup and bring to a boil. Simmer over moderately high heat until reduced to ¼ cup, about 3 minutes. Add the stock and simmer until reduced to ½ cup, about 5 minutes; discard the thyme sprigs. Whisk in the butter and season lightly with salt; keep warm.

On a large rimmed baking sheet, toss the carrots with the apples, orange juice and olive oil; season with salt. Roast in the lower third of the oven, stirring once, until tender and browned in spots, about 25 minutes.

Meanwhile, drain the pork tenderloins and cut them in half crosswise; discard the brine. Pat the pork dry and season lightly with salt. In a large cast-iron skillet, heat 1 tablespoon of the canola oil until shimmering. Add half the pork and cook over moderately high heat, turning, until browned all over, about 10 minutes. Transfer the pork to a rimmed baking sheet. Wipe out the skillet and repeat with the remaining 1 tablespoon of canola oil and pork. Roast the pork in the upper third of the oven for about 12 minutes, until an instant-read thermometer inserted in the thickest part of the meat registers 140°. Transfer the pork to a work surface and let rest for 10 minutes.

Slice the pork and transfer to plates. Drizzle the sauce on top and serve with the roasted carrots and apples.