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Before you begin, wash and dry the leeks and scallions and finely chop the greens so that you have 8 ounces/225 g of leek greens and 4 1/4 ounces/120 g of scallion greens.
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Melt butter and oil in a large Dutch oven over low heat. Add leek and scallion greens, stirring well to coat. Cook over low heat, stirring often, for about 1 hour or until they are very soft and just beginning to brown. Browning will bring out the naturally sweet flavors, but over-browning will ruin the dish, so take care.
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Once the leek and scallion greens are very soft - the green mixture might look a bit pasty, which is OK - add the beef stock, parsley, thyme and bay leaf and bring the soup to a simmer, covered, for about 10 minutes. Fish out the parsley, thyme and bay leaf, taste, and season with salt and pepper, if desired.
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Warm your soup bowls by rinsing them with hot water while you toast your bread. Preheat broiler to high with rack about 4 inches (10 cm) below.
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Place the bowls on a rimmed sheet pan. Ladle hot soup into bowls, place a slice of bread in each bowl and divide the cheese amongst the bowls. Place under broiler until cheese is melted and bubbling, about 1 to 4 minutes. Serve immediately.