Start by setting your cooker to Sauté and adjusting to 30 minutes at 390 degrees F (200 degrees C). Add Olive Oil (1 tablespoon) and half of the Country-Style Beef Ribs (0.75 pound) Stir to sear and give the meat color, 1-2 minutes a side, then remove.
Add the other half of Country-Style Beef Ribs (0.75 pound) and stir to sear, then re-add the rest of the beef. Season with Salt and Pepper (to taste). Add the Water (4 1/2 cup), Beef Bouillon Cube (2) and stir, cover the cooker and switch the setting to the stew function for 15 minutes.
In another pot sauté Olive Oil (1 tablespoon) and Butter (3 tablespoon). Add the Yellow Onion (1.5 pound) and let it sit for 6 minutes.
Let the onions sauté for another 15 minutes, but stir every 5 minutes.
Add the All-Purpose Flour (3 tablespoon) and stir. Pour in the beef rib and broth. Add Brown Sugar (2 teaspoon), Dried Thyme (1 1/2 teaspoon), Balsamic Vinegar (1 tablespoon), Salt and Pepper (to taste). Cover with lid and cook in pressure cooker for 5 minutes using the soup function.
Meanwhile, preheat oven at 400 degrees F (205 degrees C). Slice the French Bread (to taste), drizzle Olive Oil (1 tablespoon) over the top, and place in oven for about 10 minutes to toast.
Release the pressure on the cooker. Open and stir the soup.
Serve in oven-safe bowls, top with a slice of French bread and Mozzarella Cheese (to taste) then place in oven on broil on a baking tray for about 5 minutes.