GETTING READY
1 In a pan, heat the stock.
2 Preheat the oven to 320°F.
3 On a cookie tray, arrange the bread slices and bake for 30 minutes to make them dry and crisp.
4 Cut the onions in half from top to bottom.
5 Slice very thinly, also from top to bottom.

MAKING
6 In a large, heavy bottomed saucepan, heat butter with 1 ½ tbsp olive oil.
7 Sweat the onions over low heat for 15 minutes, with the lid on.
8 Sprinkle with 1 tsp salt and a generous pinch of sugar.
9 Continue cooking over low heat, uncovered, stirring frequently.
10 When the onions are golden brown, add the flour, and stir well with a wooden spoon while cooking for about 2 minutes.
11 Remove the saucepan from the heat.
12 Gradually add in the boiling stock, beating continuously and vigorously with a balloon whisk.
13 Add the vermouth or white wine and stir.
14 Return the saucepan to a low heat and simmer for 40 minutes.
15 Periodically skim off any skin that may form on the surface.
16 Add a little more salt if wished and the brandy.
17 In a bowl, mix the 3/4 cup of Parmesan cheese with the 1/2 cup grated Swiss cheese and set aside.
18 Pour the soup into hot, ovenproof earthenware bowls or heatproof china bowls.
19 Sprinkle the Swiss cheese flakes and the grated raw onion evenly over the surface.
20 Place 2 crisped bread croutons in each bowl of soup and sprinkle them with the grated cheese.
21 Drizzle a little olive oil over the cheese and add some black pepper.
22 Bake for 20 minutes.
23 Make sure the broiler is very hot and place the bowls under it for the last 5 minutes to make the cheese topping crunchy and golden brown.

SERVING
24 Serve at once.