Preheat oven to 325ºF.
Season the 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, add the short ribs and cook until caramelized on all sides. Transfer to a plate and set aside.
In the same pot, add sliced onions, reduce the 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 the flour to coat the ingredients before pouring in beef broth and cooking sherry; scrape any bits off the bottom. Stir in tomato paste, rosemary, balsamic vinegar, and cayenne pepper.
Bring it back to a simmer on medium-high. Return the short ribs and pour more broth over, enough to cover the meat.
Cover and transfer to the preheated oven. Braise for 2 hours or until the ribs are fork-tender.
Remove from the heat and transfer the short ribs to a plate while working on the sauce.
Put the heat on medium and skim off the fat on the surface.
When simmering, add the cornstarch mixture and stir until the sauce thickens, less than a minute. Before serving, taste and adjust the seasoning if necessary.
Place short ribs over mashed potato, ladle on some sauce and sprinkle fresh chopped parsley on top.