Preheat the oven to 350°F (175°C, or gas mark 4).
Soak the porcini mushrooms in warm water for about 15 to 20 minutes. Save about 1⁄3 cup (80 ml) of the water from the mushrooms and discard the rest. Use your hands to squeeze the excess liquid out of the rehydrated mushrooms, cut the larger pieces into two and set them aside.
Pat the short ribs dry and season with salt and pepper on both sides.
Heat the ghee in a large Dutch oven over medium-high heat and sear the ribs for 3 to 5 minutes on all sides. Remove the ribs from the pan and set them aside. Add the onion and garlic [and salt, as we mention above, if desired] and sauté for about 3 to 5 minutes. Add the tomato paste to the pot. Stir and cook for 1 to 2 minutes. Deglaze the pan by adding the broth along with the liquid you saved from the mushrooms. Stir and lift up all the brown bits from the bottom of the pan.
Return the ribs to the pot and nestle them in the sauce along with the porcini and portobello mushrooms and bay leaf. Stir all the ingredients to combine and transfer the pot to the oven. Cover and cook for 2 hours, or until the ribs are tender and falling off the bone, stirring about halfway through cooking. Remove the ribs from the oven and skim off any extra fat from the top of the sauce.
Discard the bay leaf and transfer the ribs to a serving dish. Season the sauce with salt and pepper to taste. Allow the ribs to cool for about 5 minutes and serve with the sauce.