Quarter onions before rendering them into 1/4" pieces. Heat oil and butter together in a heavy saucepan. Fry the onions briskly for a few minutes while stirring constantly before reducing the heat and cooking gently for 45-60 minutes. As the onions begin to turn in color they will need more requent stirring.
When the onions are mahogany brown, add the stock and a bit of salt and peper. Simmer partly covered(?) for 30 minutes and season to taste.
Toast the french bread under a preheated broiler. Serve broth in ovenproof bowls and top with a piece of bread and a mound of cheese on each. Return to broiler until cheese is golden brown.