The exact recipe that Disney's suppliers use is not available. The below recipes are very close.

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SMOKED TURKEY

1 cup tender quick salt
1/2 cup brown sugar
5 tsp. liquid smoke
1 gallon water (or amount to cover pieces)
1 turkey, chicken or turkey legs

Add all together and keep in covered container in refrigerator. Let stand for 24 hours.

When ready to roast, rinse off turkey and roast 2 hours at 350 degrees (whole bird) and then lower oven to 300 degrees until turkey is done. Turkey legs, pork chops, steaks and roasts can also be used this way. Adjust roast times for different cuts.

NOTE: These would also be delicious made in a smoker or slowly grilled over indirect heat.
Source: Uncle Phadreus

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ANOTHER BRINE RECIPE

1 cup of Tender Quick
1 cup of Table Salt
10 1/2 TBSP of liquid smoke
2 gallons of water

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TENDER QUICK INFORMATION
Tender Quick is a blend of salt, sugar, sodium nitrate, sodium nitrite, and propylene glycol. The amount of sodium nitrate and sodium nitrite are 0.5%.

A few alternate recipes -

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GRILLED TURKEY LEGS
Makes 6 servings

1 (2 liter) bottle lemon-lime flavored carbonated beverage
2 tablespoons sugar
2 tablespoons hot sauce
1 tablespoon crushed red pepper flakes
1 tablespoon black pepper
1 large sweet onion, sliced
4 turkey legs
2 tablespoons honey
1 tablespoon steak seasoning

Preheat an outdoor grill for high heat, and lightly oil grate.

In a large pot, mix the lemon-lime flavored carbonated beverage, sugar, hot sauce, red pepper, pepper, and onion. Place the turkey legs in the mixture, and bring to a boil.

Cook 30 to 45 minutes, until the turkey has reached an internal temperature of 180 degrees F (80 degrees C).

Remove onion slices from the mixture, and arrange on the prepared grill. Place turkey legs over the onions. Drizzle with honey, and season with steak seasoning.

Cook, tuning once, 20 minutes, or until a crisp browned crust has formed on the turkey legs.

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BRINED SMOKED TURKEY

FOR BRINE:
1 cup salt
1 cup brown sugar
2 gallons vegetable stock
2 cups apple cider vinegar
3 tablespoons pickling spice
1 onion, quartered
1 carrot, chopped into 2-inch pieces
1 tablespoon whole peppercorns
4 bay leaves
1 (10 pound) bag of ice
2 gallons cold water

16 to 18 pound turkey
2 tablespoons Worcestershire sauce
4 tablespoon kitchen bouquet browning sauce
2 tablespoons salt
Ground black pepper, to taste
2 tablespoons paprika
6 large oranges, quartered

Mix salt, brown sugar, vegetable stock, apple cider vinegar, pickling spice, onion, carrot, peppercorns and bay leaves in a stockpot and bring to a boil until sugar and salt is well dissolved. Remove from heat and cool.

In large pot or ice chest, add stockpot mixture to ice and cold water. Submerge turkey. Brine for 4 to 6 hours. Remove turkey from brine and dry well.

Combine Worcestershire sauce and kitchen bouquet browning sauce together and brush on skin of turkey. Rub salt, pepper and paprika onto turkey skin. Stuff inside of turkey with orange quarters and tie legs together.

Smoke turkey in smoker set to 250 degrees for about 7 hours. Checking internal temperature to be 170 degrees. Remove and let rest for 45 minutes.