* Sugar helps the onions brown. ** At least 2 cups of the beef stock should be hot. This is certainly one of the all-time favorites. The canned and packaged onion soups I've tried have been very good, and the only way to better them is for you to have your own beautiful beef stock. However, bought onion soup will benefit from the flavor additions suggested here.

Timing: For most delicious results, you want a slow simmer for 2 3/4 to 3 hours.

Special Equipment Suggested:

A food processor with slicing blade or a hand slicer is useful for the onions; a heavy-bottomed 3-quart saucepan with cover for onion cooking and simmering.

Browning the onions - 40 minutes. Set the saucepan over moderate heat with the butter and oil; when the butter has melt- ed, stir in the onions, cover the pan, and cook slowly until tender and translucent, about 10 minutes. Blend in the salt and sugar, raise heat to moderately high, and let the onions brown, stirring frequently until they are dark walnut color, 25 to 30 minutes.

Simmering the soup. Sprinkle in the flour and cook slowly, stirring, for another 3 to 4 minutes. Remove from heat, let cool a moment, then whisk in 2 cups of hot stock. When well blended, bring to the simmer, adding the rest of the stock, the Cognac or brandy, and the vermouth. Cover loosely, and simmer very slowly 1 1/2 hours, adding a little water if the liquid reduces too much. Correct seasoning.

Ahead-of-time note: May be prepared in advance; chill uncovered, then cover and refrigerate or freeze.

Serving. Serve the soup as it is, accompanying it with French bread and a bowl of grated Swiss or Parmesan cheese.