Set saucepan over moderate heat with butter and oil; when butter has melted, stir in onions, cover and cook slowly until onions are translucent and tender, about 10 minutes.
Blend in salt and sugar and raise heat to medium-high, let the onions brown, stirring often until they are a dark walnut color, 25-30 minutes.
Sprinkle flour over onions and cook slowly, stirring for another 3-4 minutes.
Remove from heat, let cool a moment, then whisk in 2 cups of hot stock. When blended, bring to a simmer, adding rest of stock, the brandy and Vermouth.
Cover loosely, and simmer very slowly for 1 1/2 hours, adding a little water if the liquid reduces too much.
Correct seasonings. May be served with some freshly grated Swiss and Parmesan cheese over top.
Soup may also be served with toasted french bread slice and cheeses. Ladle soup into individual heat proof ramekins. Top with toasted french bread slice, 1 slice Provolone, 1 slice Swiss cheese and grated fresh Parmesan cheese. Put under broiler until golden and bubbly. Bon Apetite!
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