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1
Put the wine in a wide saucepan and add the shallots, garlic, carrots, bay leaf, thyme and parsley sprigs, peppercorns and honey.
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2
Cut off the stems of the mushrooms and add them.
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3
Set the caps aside.
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4
Bring to the simmer and cook 15 minutes.
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5
Strain the mixture and discard the solids.
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6
Return the wine liquid to the saucepan.
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7
Heat one tablespoon of the butter in a skillet large enough to hold the mushroom caps in one layer when they are added.
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8
Add the mushrooms cap-side down.
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9
Cook two to three minutes on one side or until golden brown.
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10
Turn and continue cooking two to three minutes.
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11
Remove from the heat.
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12
Break each egg into a small bowl.
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13
It is best to poach only four eggs at a time.
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14
Bring the wine liquid to the simmer and carefully slide the eggs, one at a time, into the liquid.
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15
Let simmer until the whites are firm and the yolks remain runny, about two to three minutes.
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16
Carefully scoop out each egg, using a slotted spoon, and drain on absorbent paper towels.
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17
Repeat with the remaining eggs and drain.
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18
Arrange the croutons on a serving platter or arrange two croutons on each of four serving dishes.
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19
Top each crouton with a poached egg.
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20
Meanwhile, knead the remaining four tablespoons of butter with the flour.
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21
Skim off and discard any egg remnants floating on the surface of the wine liquid.
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22
Bring the liquid to the simmer and gradually add the butter-flour mixture.
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23
Stir until the sauce is thickened.
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24
Pour the sauce over the eggs on croutons and garnish with the mushrooms on the side.
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25
Sprinkle each egg with chopped parsley and serve.