Step 1

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.

Step 2

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 1/4 cup, about 3 minutes. Add the stock and simmer until reduced to 1/2 cup, about 5 minutes; discard the thyme sprigs. Whisk in the butter and season lightly with salt; keep warm.

Step 3

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.

Step 4

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.

Step 5

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