The night before cooking, brine the bird: Combine the hot water, salt, and brown sugar in a 5-gallon upright drink cooler. Stir to dissolve the salt and sugar. Add the ice and stir until the mixture is cool. (There should still be visible pieces of ice in the water.) Gently lower the turkey into the cooler, breast-end down. Weigh down if necessary to keep the bird submerged. Cover the cooler and set in a cool place for 12-18 hours.
When you're ready to roast, heat the oven to 500 F and adjust the oven racks so that the turkey will fit on the rack on the second lowest level in the oven. Spread the carrots, leek, mushrooms, and rosemary in a single layer on a half sheet pan and top with an oven-safe wire cooling rack.
Remove the turkey from the brine and rinse inside and out with cold water. Discard the brine and pat the bird as dry as you can with paper towels. Place the bird on the wire rack above the veg-rosemary mixture and tuck the wings underneath the bird.
Pre-form an aluminum shield (aka a turkey triangle) by folding a large sheet of heavy duty aluminum foil to form a triangle. Place over the turkey breast so that the point is toward the legs. Press on the sides of the foil to form it to the breast and hold its shape. Remove and set aside next to the oven for now. Brush the turkey lightly with oil.
Roast in the oven for 30 minutes. (If you have an exhaust fan, turn it on. If you don't, open a window; there will be some smoke.) Carefully remove the turkey from the oven and, working quickly, insert a probe thermometer into the thickest part of the breast, avoiding any bones. If the breast skin is already very dark, top with the turkey triangle, If it is golden brown, don't add the turkey triangle, but keep an eye on the skin and add it if it is getting too dark.
Return the turkey to the oven and reduce the oven temperature to 350 F. Set an alarm on the probe to go off at 155 F. A 14-pound turkey should take an additional 1 1/2 to 2 hours, but the temperature of the meat is more important than the time it takes to get there, so keep an eye on the thermometer.
When the alarm goes off, transfer the turkey to a second cooling rack on a second half sheet pan and leave the bird to rest for 45 minutes before transferring to a cutting board for carving.
Meanwhile make the gravy: Remove the wire rack from the roasting pan and put the rack in the sink. Use a metal spatula to scrape up all of the vegetables and drippings from the bottom of the pan, and transfer to a 4-quart saucepan or saucier. (If the sheet pan is dry, pour enough of the broth to coat the bottom of the sheet pan, then place over medium-high heat and bring the broth to a simmer. Use a wooden spoon or spatula to dislodge the turkey drippings from the bottom of the sheet pan. You'll need to rotate the pan around on the heat-use an oven mitt-as you go. Carefully pour into the saucepan.)
Add the rest of the broth to the saucepan, place over medium-high heat, and bring to a simmer. Reduce the heat to medium-low and hold the simmer until the broth is flavorful, about 10 minutes.
Strain the broth through a fine-mesh sieve set over a large heatproof measuring cup or bowl, pressing on the solids to extract as much liquid as possible. Discard the solids. Remove most of the fat using a ladle or fat separator. Measure out 4 cups of the fortified broth, saving any remainder for another use. Wipe out the saucepan.
Add the butter to the now empty saucepan and return to medium heat to melt. Once the butter has melted, briskly whisk in the flour, followed by the thyme. Continue whisking until the flour and fat (the "roux") lighten in color and foam, about 1 minute. Then, still whisking, slowly pour in the 4 cups fortified broth. Bring to a simmer, then reduce the heat to medium-low and continue to cook, whisking frequently, until thickened, about 7 minutes. Remove from the heat and stir in the vinegar and pepper. Taste and season with additional pepper and/or salt. Serve with the turkey.