In a large saucepan, heat 2 Tablespoons salted butter and the olive oil together over medium heat. Once the oil and butter are melted, add the onions, stirring occasionally for 1 hour. Make sure the onions are completely cooked and caramelized. They should be gold brown.

You can add a dash of water or Guinness to deglaze the saucepan and prevent burning.

Gently stir in the brown sugar and thyme. Mix for a minute and add the garlic. Cook for 1 more minute. Add another 2 Tablespoons salted butter and allow it to melt over low heat. Sprinkle the flour on top, stir gently and cook for 4 minutes, stirring continuously.

Slowly, over medium heat, stir in the Guinness and deglaze the pan. Stir in the broth, Worcestershire sauce, grainy mustard, bay leaves, salt, and pepper.

Slowly bring the soup to a boil. Then reduce the heat and simmer for 30 minutes.

While the soup reduces, place oven-proof bowls on a rimmed baking sheet. When the soup is ready, ladle the soup into the bowls. Top each bowl with a slice of bread and sprinkle with the white cheddar cheese.

Put the tray in the oven and broil until the soup melts. About 2 minutes. Serve immediately.