Press the Sauté button on the Instant Pot® and melt 3 tablespoons butter. Add onions and brown sugar. Cook until onions are just tender, about 5 minutes. Press the Cancel button. Close lid, set steam release to Sealing, press the Manual button, and adjust time to 15 minutes. When the timer beeps, quick-release the pressure and press the Cancel
Drain off the excess liquid from pot. Return pot to machine and press the Sauté button. Add remaining butter, and cook, stirring frequently, until onions are caramelized and brown, about 15–20 minutes.
Add wine to pot and scrape up any browned bits from pot. Press the Cancel button. Add garlic, vinegar, bay leaf, thyme, pepper, stock, and broth to pot. Close lid, set steam release to Sealing, press the Manual button, and set time to 20 minutes. When the timer beeps, let pressure release naturally, about 20 minutes. Press the Cancel button, open lid, and stir well. Remove and discard bay leaf.
Press the Sauté button and simmer soup for 10 minutes, stirring often, to thicken. While soup simmers, heat oven broiler.
Divide soup into heatproof soup bowls. Place a slice of bread on each bowl, then divide cheese over bowls. Broil for 3–5 minutes, or until cheese is bubbling and brown. Serve hot.