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For the turkey: The day before serving, heat 3 quarts of the water with salt and sugar in a large pan, stirring to dissolve salt and sugar.
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Remove from heat, add remaining 1 quart cold water, and stir.
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Set brine aside to cool to room temperature.
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Put turkey in a large container and cover with the brine.
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Cover and refrigerate for 4 to 5 hours.
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(If you want to brine the turkey overnight, use half the amount of salt and sugar.)
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For rub: Crumble annatto paste into a food processor and blend with the garlic, oregano, coriander, black pepper, salt, and allspice.
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Add the fruit juices and the olive oil and process to make a pasty sauce.
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Drain and pat the turkey very dry.
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Smear the rub all over the bird.
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Cover and refrigerate overnight.
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Prepare an outdoor grill with a large medium-heat fire for direct and indirect grilling.
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Position a drip pan under the grate on indirect side of the grill.
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Place turkey, breast side up, over the drip pan and grill, covered, until meat is cooked about halfway through, about 50 minutes.
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Turn and cook until an instant-read thermometer inserted in the thigh registers 170 degrees F, about 50 minutes more.
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Move turkey to direct heat and rotate to evenly brown the skin in the last 10 minutes of cooking.
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Transfer to a carving board, cover, and let rest 10 minutes before carving.
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Shopsmart: Annatto paste is a Mexican flavoring composed of ground annatto, or achiote, seeds mixed with herbs-usually oregano-and spices.
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The rust-color paste is then pressed into a compact brick.
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Look for it in the Hispanic section of your grocery.
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Cook's Note: The turkey can be grilled as soon as it's rubbed with the spice paste, and it will be great.
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But if you have time, let it marinate-it will be awesome.