Combine flour, salt, and pepper in a large resealable plastic bag. Add pork chops and shake well to coat thoroughly.

Melt butter in a large skillet over medium-high heat. Add olive oil and heat until sizzling. Add pork chops in a single layer. Saute until browned on one side, 1 1/2 to 2 minutes. Turn pork chops over and cook for an additional 1 1/2 to 2 minutes. Transfer pork chops to a plate.

Add beer, vinegar, mustard, brown sugar, and rosemary to the skillet and mix well. Bring to a boil over medium-high heat. Return pork chops to the skillet, cover, and reduce heat to medium-low. Simmer until pork chops are no longer pink in the center, 10 to 12 minutes. An instant-read thermometer inserted into the center should read 145 degrees F (63 degrees C).

Transfer pork chops to a plate. Increase heat to medium and boil pan juices in the skillet until thickened, whisking often, about 3 minutes. Top pork chops with sauce.