In a small bowl, combine the flour, 1 tsp. salt, 1/2 tsp. pepper and the allspice. Coat the chicken with the flour mixture. In a large, heavy skillet, heat 1 tbsp. butter over medium-high heat. Add the chicken skin side down and cook until browned, about 5 minutes. Flip and cook until browned on the other side, 3 to 5 minutes; transfer to a plate. Discard all but 1 tbsp. fat from the skillet. Add the chopped onion to the skillet and cook until softened, 3 minutes. Stir in the beer, scraping up any browned bits.
Return the chicken to the skillet, lower the heat, cover and simmer until cooked through, about 20 minutes. Transfer to a platter; tent with foil. Boil the liquid in the skillet for 2 minutes to reduce slightly. Whisk in 2 tbsp. butter and the parsley.
Meanwhile, in another large skillet, heat the remaining 1 tbsp. butter over medium heat. Add the sliced onion and cook until softened, about 3 minutes. Stir in the carrots, cover and cook for 5 minutes. Add the green beans and 3 tbsp. water, cover and cook until crisp-tender, about 5 minutes; season with salt and pepper. Divide the vegetables among 4 plates and top with the chicken and sauce.
