Preheat oven to 300F. Pat chuck roast dry and season all over with salt and pepper. Heat a Dutch oven (or a large oven-safe high-walled skillet) over medium-high heat, then add butter and olive oil. Sear chuck roast 2-3 minutes on all sides until a deep golden brown. Transfer to a plate and set aside.
In remaining drippings over medium heat, saute carrot, onion, and mushrooms until vegetables begin to soften, about 3 minutes. Add garlic and saute 1 additional minute until fragrant. Pour beer and beef broth into pan and bring to a simmer.
Nestle chuck roast back into Dutch oven and arrange bay leaf, rosemary, and thyme around roast. Cover Dutch oven and place in 300F oven and let cook 2-1/2 to 3 hours or until chuck roast is very tender.
Remove chuck roast and vegetables from Dutch oven and transfer to a plate. Shred roast into bite-sized pieces. Remove and discard bay leaf and thyme/rosemary sprigs.
Whisk flour with a little bit of braising liquid from Dutch oven until smooth. Add flour mixture back to pan, stirring until smooth. Bring to a simmer over medium-high heat until thickened. Return beef and vegetables back to Dutch oven. Serve warm with mashed potatoes.