In a small bowl, whisk together ketchup, brown sugar, vinegar, Worcestershire, and paprika. Season with salt and pepper.

Dice onion and garlic into a second small bowl.

Heat 1 tablespoon oil in a 5-quart Dutch oven or other heavy pot over medium-high. Season chicken generously with salt and pepper. Working in two or three batches, and adding remaining 1 tablespoon oil as necessary, sear chicken until golden, about 4 minutes per side. Transfer to a plate.

Drain all but 1 tablespoon oil from Dutch oven; reduce heat to medium. Add onion and garlic; deglaze, stirring, until tender, 6 to 8 minutes. Add beer and barbecue sauce; stir until combined.

Return chicken and any accumulated juices to Dutch oven. Bring to a boil, then reduce heat, cover, and simmer until chicken is very tender, about 1 hour. Turn off heat.

Transfer chicken to plate; let cool. When cool enough to handle, remove skins and bones; discard bones. Shred chicken into large pieces with two forks.

Bring sauce to a simmer and reduce until slightly thickened, about 30 minutes. (You should have about 3 cups.) Return shredded chicken to Dutch oven and heat through, or let cool to room temperature and refrigerate in an airtight container up to 3 days (gently reheat before serving). Serve on potato buns.