1. Melt butter with oil in a large Dutch oven over medium heat. Add onions and sugar. Cook, stirring often, 45 to 50 minutes or until onions are caramel colored.
2. Sprinkle onions with flour, and stir to coat. Add chicken broth. Bring to a boil over medium heat, and cook 20 minutes. Add hash browns and 2 cups water. Reduce heat to low, add thyme and next 3 ingredients, and simmer 30 minutes.
3. Stir in cheese and half-and-half; cook, stirring constantly, over medium heat 5 minutes or until cheese is melted. Remove bay leaf before serving. Garnish, if desired.