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Quarter eggs lengthwise.
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Arrange in a single layer in broiler-safe 2 qt round dish.
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Set aside.
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Remove green portion from leeks, and halve lengthwise.
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Wash thoroughly, and pat dry.
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Remove the roots, and cut the leeks lengthwise into thin strips (julienne).
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Heat 4 tbs of the butter in a large skillet over medium heat, add leek strips.
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Cook leeks until tender, stirring occasionally.
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Spoon leeks over eggs in dish.
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Melt remaining 2 tbs butter in a skillet.
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Whisk in flour.
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Add milk, cook until thickened and bubbly.
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Whisk in 1/4 cup of the white cheddar cheese, the Parmesan cheese, and the cayenne pepper.
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Season to taste with salt and pepper.
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Pour the sauce over the leeks and eggs in dish.
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Sprinkle with remaining white cheddar cheese.
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Bake uncovered about 10 minutes in 450 oven, or until the edges are bubbly and the top begins to brown.