Preparation

To make the Cajun spice mixture:

In a small bowl, mix all the ingredients to blend.

To brine the turkey:

In a large pot, bring 1 quart of water to a boil over high heat. Add the beer, salt, sugar, and 1/3 cup of the spice mixture and stir until the sugar has dissolved. Add 2 rosemary sprigs and remove from the heat.

Pour the hot brine into a container large enough to hold the turkey and add the remaining 5 quarts of cold water to cool the brine. Place the turkey into the brine, making sure it is entirely submerged. Cover the container tightly with a lid or plastic wrap and refrigerate for about 12 hours or overnight.

To roast the turkey:

Preheat the oven to 350˚F.

Remove the turkey from the brine and discard the brine, reserving the rosemary sprigs, and then pat the turkey skin dry with a clean towel. Stuff the main turkey cavity with half of the chopped celery, bell pepper, onion, and reserved brined rosemary sprigs, and tie the turkey legs together with butcher’s twine.

In a small heavy saucepan, melt 1/2 cup (1 stick) of the butter. Remove the pan from the heat and stir in 2 tablespoons of the spice mixture.

Place the remaining celery, bell pepper, and onion in a large roasting pan with a rack. Set the roasting rack in the pan and place the turkey on the rack. Brush some of the spiced butter all over the turkey.

Cover the pan with aluminum foil and roast the turkey for 1 hour and 40 minutes. Baste the turkey and continue roasting uncovered for about 1 hour and 20 minutes longer, or until a meat thermometer reads 160°F when inserted into the part of the thigh nearest to the thigh and hip joint.

Transfer the turkey to a carving board (do not clean out the roasting pan), and rest at room temperature for 30 minutes before carving.

Meanwhile, to make the gravy:

Set the roasting pan on the stove over medium-high heat. Add the chicken stock, thyme sprigs, and remaining rosemary sprig to the pan drippings in the roasting pan, and bring the mixture to a simmer, stirring often to scrape up the brown bits. Strain the pan juices, and discard all the solids. Spoon off the fat that has settled to the top of the pan juices.

In a medium heavy saucepan, melt the remaining 1/4 cup (1/2 stick) of butter over medium heat. Stir in the remaining 2 teaspoons of the spice mixture (reserve any extra spice mixture for another use) and cook for about 30 seconds, or until fragrant and toasted. Stir in the flour and cook, stirring often, for about 2 minutes. Whisk in the pan juices and bring the mixture to a simmer over medium-high heat. Simmer for about 5 minutes, or until the gravy thickens slightly. Season the gravy to taste with salt and pepper.