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1
Put the eggs in a medium saucepan and cover with hot tap water.
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2
Bring to a boil over high heat, then lower the heat to moderate and simmer for 10 minutes.
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3
Drain off the water and shake the pan to lightly crack the eggs.
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4
Add ice and cold water to the pan and let stand until the eggs are cold.
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5
Drain and peel the eggs.
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6
Meanwhile, in a medium skillet, melt 1 tablespoon of the butter over moderately high heat.
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7
When the foam subsides, ad the scallions and cook until just wilted.
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8
Add the mushrooms and cook, stirring occasionally, until all the liquid is evaporated and the mushrooms begin to brown, about 10 minutes.
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9
Stir in the anchovy and season with salt and pepper.
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10
Spread the mushroom mixture on the bottom of a 6-cup gratin dish.
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11
Slice the eggs with an egg slicer, arrange them over the mushrooms in an even layer and season lightly with salt and pepper.
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12
Preheat the broiler.
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13
Wipe out the skillet and melt 1 teaspoon of the butter.
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14
Add the onions and 1/2 cup of water and cook over moderately low heat until they are tender and the water is almost evaporated, about 12 minutes.
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15
Transfer the onions to a food processor and puree.
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16
In a medium saucepan, melt the remaining butter.
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17
Whisk in the flour until blended and cook for 15 seconds.
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18
Add the milk and bring to a boil, whisking frequently.
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19
When the sauce thickens, add the cream and the onion puree, season with salt and pepper and bring to a boil.
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20
Pour the sauce over the eggs, shaking the dish gently to let the sauce seep down.
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21
Sprinkle the cheese on top and broil for 3 to 4 minutes, until browned and bubbling.
