Rinse the chicken and pat dry. Toss the ginger with 2 tablespoons salt and 1 teaspoon pepper. Sprinkle half of the mixture on the chicken; reserve the rest. Refrigerate the chicken, uncovered, at least 2 hours.
Meanwhile, make the sauce: Pour the ginger beer into a wide skillet; cover and bring to a boil. Uncover and cook over high heat until reduced by half, about 30 minutes. Remove from the heat, add the tea bags and steep 10 minutes; discard the tea bags. Add the onion, tomatoes, mustard, Worcestershire sauce, brown sugar and thyme. Return to high heat and cook, stirring occasionally, until thick, about 50 minutes, using a potato masher to crush the tomatoes during the last 5 minutes. Remove the thyme. Set aside half of the sauce for serving.
Bring the chicken to room temperature about 30 minutes before grilling. Preheat a grill to medium with indirect heat: On a charcoal grill, bank the coals to one side; on a gas grill, turn off half of the burners.
Brush the chicken with some of the butter and sprinkle with some of the remaining ground ginger mixture. Place skin-side up on the cooler side of the grill; cover and cook until it starts browning, about 18 minutes, basting with more butter and sprinkling with more ground ginger mixture halfway through. Brush the chicken with a thick layer of sauce. Cover and cook until a thermometer inserted into the thickest part registers 160°F, about 20 minutes, basting with more sauce halfway through.
Brush the chicken with more butter, then turn skin-side down and place on the hotter side of the grill (over direct heat). Cook, uncovered, until marked, about 2 minutes. Rotate the chicken to make crisscross grill marks and cook 2 more minutes. Transfer to a board and let rest about 10 minutes. Season with salt and serve with the reserved sauce.