In saucepan place beef stock and reduce by 50% over medium low heat.
In a large fry pan melt butter. Add onions, salt and pepper cooking until onions are soft. Add carrots and celery, cook 3 minutes. Add tomato sauce and cook 3 more minutes.
Add flour and cook rue until it is a light peanut butter color. Take care not to scorch, stirring rue constantly.
Add reduced stock to the vegetables. Continue to cook sauce and allow to reduce.
Strain sauce into saucepan and add Bouquet Garni. Let sauce simmer until it coats the back of a spoon. Season with salt and pepper. The result is Brown Sauce.
To make Demi Glace, simply put equal parts of Brown Sauce and Beef Stock in a saucepan. Reduce volume by half.
Chill sauce and place in refrigerator or freeze for use later.