Season short ribs all over with salt and black pepper.
Heat vegetable oil in a skillet over medium-high heat. Cook short ribs in hot oil until browned on all sides, 7 to 12 minutes. Transfer ribs to a sprayed Dutch oven.
Return skillet to heat and saute onion with a pinch of salt in hot pan until softened, about 3 minutes. Add garlic and saute until fragrant, about 1 minute more. Stir mushrooms into onion mixture.
Pour sauce liquid into skillet and bring to a boil while scraping the browned bits of food off of the bottom of the pan with a wooden spoon. Stir beef broth, tomato sauce, rosemary, 1/2 teaspoon salt, cayenne pepper, and bay leaf into onion mixture.
Pour tomato mixture over short ribs into Dutch oven and cover Dutch oven with a lid.
Cook short ribs in the preheated oven until short ribs are fork-tender, about 2 hours.