Preheat oven to 325ºF. Season short ribs with salt and pepper on all sides; set aside. In a large Dutch oven over medium-high heat, add oil and butter. When hot, place short ribs until caramelized on all sides. Transfer on a plate and set aside.
In the same pot, add onion, reduce heat to medium and sauté until soft, about 5 minutes. Add shiitake mushrooms and cook for 3 minutes, stirring constantly to prevent from burning. Add garlic and sauté for 1 minute. Stir in flour to coat ingredients. Pour in beef broth and cooking sherry; scrap any bits at the bottom. Stir in tomato paste, rosemary, balsamic vinegar and cayenne.
Bring it back to a simmer on medium-high. Place short ribs back and pour more broth over enough to cover meat. Cover and transfer to the preheated oven. Braise for 2 hours or until ribs are fork-tender.
Remove from heat and transfer short ribs on a plate while working on sauce.
Put heat on medium and skim off fat on the surface. When simmering, pour in cornstarch mixture and stir until sauce thickens, less than a minute. Taste and adjust seasoning if necessary. Place short ribs over mashed potato, ladle sauce over and sprinkle fresh parsley on top. Makes 4 servings.