Place the butter or oil in a pot over medium heat. Add the onions and cook, stirring occasionally, until caramel-like and sticky, about 10 minutes. Stir in the garlic and cook a few seconds more. Mix in the mustard, wine, stock and thyme or tarragon; bring the soup to a gentle simmer. Simmer 20 minutes, and then season with salt and pepper.
While the soup simmers, preheat the oven to 375 degrees F. Set the bread slices on a small baking sheet. Bake 8 minutes, and then turn each slice over. Divide and top the bread with the cheese. Bake 5 minutes more, or until the cheese is melted.
Divide the soup between 2 wide bowls. Top each with a piece of cheese-topped bread, and then sprinkle with parsley and serve.