Trim short ribs, but leave some of the fat. Fat is flavor.
Preheat oven to 325 degrees F (165 degrees C).
Season short ribs with salt and black pepper.
Heat olive oil in a dutch oven over medium-high heat. Brown short ribs in hot oil until browned on all sides, 7 to 12 minutes. Remove ribs from dutch oven
Add onion with a pinch of salt to the hot dutch oven and saute until softened, about 3 minutes.
Add garlic and saute until fragrant, about 1 minute more.
Stir mushrooms and carrotis into onion mixture.
Pour half a cup of beef stock into the dutch oven and bring to a boil while scraping the browned bits of food off of the bottom of the pan with a wooden spoon. Stir remaining beef broth, tomato sauce, rosemary, thyme, 1/2 teaspoon salt, cayenne pepper, smoked paprika, and bay leaf into the pot.
Add short ribs into Dutch oven and stir make sure they are covered. Put the lid on the Dutch oven.
Put Dutch oven in the preheated oven and cook short ribs until short ribs are fork-tender, about 2 hours.
Serve over polenta.