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To Make the Broth:
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Put all of the ingredients except the water and salt into a large stockpot.
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Add the water, pouring in enough to cover the ingredients by about 2 inches.
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Bring the broth to a boil over medium-high heat, skimming any foam that forms on the surface with a skimmer.
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Reduce the heat to medium-low and simmer gently, uncovered, skimming any foam that forms on the surface during the first hour or so of cooking, for 3 to 4 hours, adding salt to taste during the last hour of cooking.
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The broth is ready when it is reduced by about one-half and has developed a rich, meaty flavor.
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Strain the broth through a colander lined with damp cheesecloth into a clean container.
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Discard the solids.
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(You can save the good, large chunks of chicken meat and large pieces of carrot and celery and refrigerate them.
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I enjoy them as a cold lunch, drizzled with olive oil and sprinkled with salt.)
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Let the broth cool to room temperature, then cover and refrigerate until well chilled.
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Skim off and discard the congealed layer of fat on the surface before reheating.
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To Make the Egg Ribbons:
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Combine the eggs, milk, parsley, salt, nutmeg, and a little pepper in a bowl and whisk together.
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Gradually whisk in the flour, taking care to avoid lumps.
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Cover the batter with plastic wrap and let stand at room temperature for 30 minutes.
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Place a 9-inch nonstick skillet or saute pan (I use a well-seasoned cast-iron skillet) over medium heat and coat the bottom with a film of butter.
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When the butter is hot, pour in a small ladleful of batter (less than 1/2 cup) and quickly swirl the pan so that the batter coats the entire bottom, forming a thin pancake.
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Cook for 30 to 45 seconds, or until just set.
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Using an offset spatula or pancake turner, flip the pancake and cook on the other side for 20 to 30 seconds, or until set.
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Transfer the pancake to a plate.
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Continue making the pancakes until you have used all the batter, taking care to add butter to the pan as needed.
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Stack the pancakes on the plate as you remove them from the skillet.
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Roll the pancakes up, one or two at a time, and cut them crosswise into thin ribbons (1/8 to 1/4 inch wide).
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Unravel the ribbons and place them on a plate or in a shallow bowl.
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Continue to roll and cut the pancakes until you have cut them all.
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Cover the ribbons with plastic wrap and set aside.
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In a saucepan, bring the broth to a boil over medium heat.
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Add the egg ribbons to the boiling broth and cook for just a few minutes, until the ribbons are heated through.
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Ladle the soup into 6 shallow rimmed bowls and sprinkle each serving with cheese.
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Serve immediately.