Place a large pot of water over high heat and bring up to a boil. Once boiling, add some salt and cook pierogies according to package directions. Just before draining, remove and reserve about 1/2 cup of the cooking liquid. Drain and reserve the pierogies.
Once you have the water on for the pierogies, preheat a large skillet over medium-high heat with 2 tablespoons of butter and the olive oil. While the skillet is heating up and the butter melting, combine the flour and paprika in a large dish. Season the chunks of chicken with salt and pepper, dredge them in the flour-paprika mixture and shake off any excess.
Add the chicken pieces to the hot skillet and spread them out in an even layer. Cook for 3 minutes without moving them around so they brown well on one side. Flip the pieces over and continue to brown for another 3 minutes. Remove the browned chicken to a plate and reserve.
Add the cabbage, celery, carrots, onion, bay leaf and some salt and pepper to the skillet and cook, stirring every now and then for 8-10 minutes, until the veggies get tender. Add the chicken back to the skillet, the pickle juice and the can o' beer. Turn the heat up to high, bring up to a bubble and simmer for 4-5 minutes, until the liquid reduces by about half.
Once everything is simmering away, place a second skillet (or the same pot you cooked the pierogies in) over medium heat. Melt the remaining 3 tablespoons of butter and continue to cook until it starts to smell nutty and is lightly brown in color. Add the reserved cooking water and mustard, and stir to combine. Add the drained pierogies, toss to coat and heat through.
To serve, divide the chicken and veggies among 4 shallow serving bowls. Top with some of the brown butter and mustard pierogies, and garnish each dish with some chopped dill and parsley.