In a heavy 4 – 5 quart saucepan, melt the butter with the oil over moderate heat. Stir in the onions and 1 tsp. salt. Cook uncovered over low heat, stirring occasionally, for 20 – 30 mins or until the
onions are a rich golden brown. Sprinkle flour over the onions and cook, stirring, for 2 to 3 mins. Remove the pan from the heat. In a
separate saucepan, bring the stock to a simmer, then stir the hot stock into the onions. Return the soup to low heat and simmer, partially covered, for another 30 – 40 minutes.
While the soup simmers, make the croutons. Preheat the oven to
325. Spread the slices of bread in one layer on a baking sheet and bake for 15 minutes. With a pastry brush, lightly coat both sides of
each slice with olive oil. Then turn the slices over and bake for another 15 minutes or until the bread is completely dry and lightly browned. Rub each slice with the cut garlic clove and set aside.
To serve, place the croutons in individual oven-safe soup bowls and ladle the soup over them. Sprinkle with cheese and bake in 350
oven for 10 to 20 minutes, or until the cheese has melted, then broil for a minute or two until the cheese has browned.