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Place broth, onion, and celery in a medium nonreactive saucepan. Cover and simmer 5 minutes until vegetables are softened. Stir in mustard and celery seeds, then stir in chili sauce, beer, bay leaf, molasses, and Worcestershire sauce. Simmer gently, uncovered, stirring often, 20 to 30 minutes until sauce is lightly thickened. Taste and season with salt, pepper, and Tabasco sauce. Discard bay leaf. Use sauce immediately, or cover and refrigerate up to a week.
Prepare a medium-hot barbecue fire. Lightly oil grill rack or coat with nonstick vegetable spray.
Use hands to pull off and discard chicken skin, then cut off and discard all visible fat. Rub all sides of chicken with oil; sprinkle lightly with salt and pepper.
Transfer half of sauce to a small bowl and set near grill for brushing. Grill chicken 3 minutes per side, then brush with some sauce and continue to grill, turning two to three times and brushing with more sauce, for an additional 10 to 12 minutes, until meat is white throughout.
Discard brushing sauce; serve chicken hot with reserved serving sauce.
Per serving: calories, 217; total fat, 3 g; sodium, 622 mg;
total carbs, 15 g; protein, 31 g; cholesterol, 75 mg.