To Make the Short Ribs
Preheat oven to 375°F.
Place large oven-proof pot over medium-high heat and add 1 tablespoon of butter. Season short ribs generously with salt and pepper and add to the pot.
Cook short ribs until browned on at least 2 sides, which can take several minutes. Short ribs are ready to be flipped when they naturally release from the bottom of the pot. If they're still sticking, give them more time. When finished browning, transfer short ribs to a plate and set aside. Drain off any excess fat from the pot.
While short ribs are browning, chop the carrot, celery, onion, mushroom and garlic. Use a knife for larger veggies and chunkier sauce. Or use a food processor to chop them smaller.
Add remaining tablespoon of butter and reduce temperature to medium-low. Add all veggies and cook until softened and slightly browned, about 5 minutes.
Add tomato paste, beer, beef broth, rosemary and bay leaf and stir until mixed with the vegetables. Add short ribs back to the pot and cover with lid or foil.
Cook short ribs for 2 - 2½ hours until meat is tender enough to pull apart with fork and knife, but not so tender that it falls off the bone. Check on it a few times, if short ribs are ever more than halfway out of the liquid, add more beef broth.
To Make The Potatoes
During the last 30 minutes of the short rib cook time, prepare the mashed potatoes.
Add potatoes to a pot of cold water and season generously with salt. Set over high heat and bring to a boil.
Reduce heat slightly and simmer until potatoes are tender, about 20 minutes. Drain.
Add butter and milk or cream to pot along with the drained potatoes and mash until smooth. Add salt and pepper and taste, adding more seasoning as needed.
Serve short ribs over mashed potatoes.