Season pork cubes with salt and black pepper.
Heat vegetable oil in a pot over high heat. Working in batches, cook and stir pork in hot oil until browned on all sides, 5 to 10 minutes. Transfer pork to a bowl and reduce heat to medium.
Melt butter in the pot. Cook and stir onion with a pinch of salt in hot butter until onion is softened and translucent, 7 to 10 minutes. Add garlic; saute until fragrant, about 30 seconds.
Stir flour into onion mixture; cook and stir until flour is completely incorporated, about 2 minutes. Add bay leaf and caraway seed; cook for 2 minutes more. Pour beer into onion mixture. Cook and stir until thickened, 1 to 3 minutes.
Stir pork, chicken broth, carrots, and celery into beer mixture; bring to simmer. Stir 1/4 cup parsley and balsamic vinegar into stew, reduce heat to medium-low, and simmer until pork is fork-tender, about 2 hours.
Bring a large pot of lightly salted water to a boil. Add the Brussels sprouts and cook uncovered until almost tender, about 5 minutes. Drain.
Stir Brussels sprouts into stew and simmer until heated through, about 5 minutes. Season with salt and pepper to taste. Divide mashed potatoes between 6 bowls. Ladle stew over potatoes and top each with a pinch of parsley.
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