For the brine and chicken: In a medium pot on high heat, add the tea, 1/4 cup salt and 1 cup water. Cook, stirring, until the salt dissolves. Remove from the heat and let cool to room temperature.
Pour the cooled brine into a large resealable plastic bag. Add the chicken thighs and breasts and refrigerate overnight.
About 2 hours before grilling, remove the chicken from the bag and pat dry. Let rest on a plate in the refrigerator for 1 1/2 hours, then at room temperature for 30 minutes.
For the sauce. In a medium pot over medium-high heat, heat the oil. Add the onions and saute until tender, about 4 minutes. Season with salt and pepper, then add the ketchup, beer, brown sugar, paprika, chile flakes, garlic powder and garlic. Cook, stirring, until the sauce thickens and the sugar is completely dissolved, about 10 minutes. Set aside.
Preheat a grill to high heat.
Salt and pepper both sides of the chicken pieces. Over the direct heat, put the chicken on the grill on what would be the skin side. Grill until the chicken easily releases and there are grill marks, 6 to 8 minutes. Flip and move the chicken to the indirect heat area of the grill and brush with the sauce. Cover and grill until cooked through, brushing with the sauce again once or twice, another 6 to 8 minutes.