Preheat oven to 350°F.
Mix the flour with a generous amount of salt and pepper on a sheet of waxed paper or on a large, rimmed baking sheet. Pat the roast dry with paper towels; dredge it in the flour mixture.
Heat butter and olive oil in a large Dutch oven over medium-high heat until it is hot but not smoking. Sear the pork on both sides (about 4-6 minutes per side), just until a nice golden-brown color develops and the meat releases easily from the pan. You don’t need to cook the meat all the way through since it will finish in the oven. Remove the pork to a plate.
Reduce the heat to medium and add the vinegar, scraping with a wooden spoon to deglaze the pan (loosening the browned bits from the bottom).
Add the carrots, onion, celery, and garlic to the pot. Cook, stirring, until the onions are translucent (about 10 minutes).
Place the pork on top of the vegetables. Add the beer, thyme, rosemary, and bay leaves. Season with salt and pepper, cover with a lid, and bake in the 350°F oven for a total of about 70-80 minutes (about 20 minutes per pound), or until the meat reaches an internal temperature of 145˚F.
Let the pork rest for 10-15 minutes before slicing and serving.