To make Cajun spice mixture: In small bowl, mix all ingredients to blend.

To brine the turkey: In large pot, bring 1 quart water to boil over high heat. Add beer, salt, sugar, and ⅓ cup of spice mixture and stir until sugar has dissolved. Add 2 rosemary sprigs and remove from heat. Pour hot brine into container large enough to hold turkey and add remaining 5 quarts of cold water to cool brine. Place turkey into brine, making sure it is entirely submerged. Cover container tightly with lid or plastic wrap and refrigerate for about 12 hours or overnight.

To roast the turkey: Preheat oven to 350˚F. Remove turkey from brine and discard brine, reserving rosemary sprigs, and then pat turkey skin dry with clean towel. Stuff main turkey cavity with half each of the celery, bell pepper, onion, and reserved brined rosemary sprigs and tie turkey legs together with butcher’s twine.

In small heavy saucepan, melt 2 tablespoons (¼ stick) of butter. Remove pan from heat and stir in 1 tablespoon of spice mixture. Place remaining celery, bell pepper, and onion in large roasting pan with rack. Set roasting rack in pan and place turkey on rack. Brush spiced butter all over turkey. Cover pan with aluminum foil and roast turkey for 1 hour and 40 minutes. Baste turkey and continue roasting uncovered for about 1 hour and 20 minutes longer, or until meat thermometer reads 160°F when inserted into part of thigh nearest to hip joint. Transfer turkey to carving board (do not clean out roasting pan), and rest at room temperature for 30 minutes before carving.

To make gravy: Set roasting pan on stove over medium-high heat. Add chicken stock, thyme sprigs, and remaining rosemary sprig to pan drippings in roasting pan, and bring mixture to simmer, stirring often to scrape up brown bits. Strain pan juices and discard all solids. Spoon off fat that settles to top of pan juices. In medium heavy saucepan, melt remaining 3 tablespoons of butter over medium heat. Stir in remaining 1 ½ teaspoons of spice mixture (reserve any extra spice mixture for another use) and cook for about 30 seconds, or until fragrant and toasted. Stir in flour and cook, stirring often, for about 2 minutes. Whisk in pan juices and bring mixture to simmer over medium-high heat. Simmer for about 5 minutes, or until gravy thickens slightly. Season gravy to taste with salt and pepper.

To serve: Carve the turkey and serve it with the gravy.