Start by heating olive oil in a large pot over medium heat. Season your beef short ribs with salt and pepper before adding them to the pot. Sear until they are browned on all sides, which should take about 4–5 minutes per side. Once browned, remove them from the pot and set aside.
Using the same pot, add your thinly sliced onions and cook them until they caramelize, stirring occasionally to prevent burning. This process typically takes around 15–20 minutes and brings out their natural sweetness.
Once your onions are perfectly caramelized, stir in minced garlic and cook for an additional 1–2 minutes until fragrant. Return your seared beef short ribs back into the pot along with any accumulated juices.
Pour in your beef broth along with balsamic vinegar, thyme, bay leaf, and additional salt and pepper if desired. Bring this mixture to a boil before reducing the heat to low; cover it up and let it simmer for about 1.5 to 2 hours until the beef becomes tender.
While waiting for your soup to finish cooking, preheat your oven broiler. Lightly toast slices of French bread on a baking sheet until they’re golden brown but not too crispy so they can absorb some soup later.
After removing the tender short ribs from the pot, shred the meat off the bones before returning it back into your soup pot while discarding any bones left behind. Ladle portions of soup into oven-safe bowls topped with toasted French bread slices followed by a generous amount of shredded cheese before broiling them for about 2–3 minutes until bubbly and golden brown on top.