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1
Preheat the oven to 225 degrees F. Place the brioche rolls and their tops on a baking sheet and place in the oven to warm.
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2
In a large skillet, heat the oil over medium-high heat.
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3
Add the ham and cook, stirring, until lightly caramelized, about 3 minutes.
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4
Add the onions, carrots, and celery and cook, stirring often for 4 minutes.
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5
Add the mushrooms and cook for 5 minutes.
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6
Add the garlic, salt, and pepper, and cook for 30 seconds.
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7
Add the chicken and cook, stirring often, until slightly caramelized, about 5 minutes.
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8
Add the stock and bring to a boil.
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9
Cook until reduced by half, about 3 minutes.
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10
Add the cream and cook until reduced by half.
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11
Add the butter to the hash and stir to incorporate.
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12
Add the thyme and Brabant potatoes, and toss to combine.
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13
Remove from the heat.
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14
Have a pot of simmering water.
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15
Add vinegar.
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16
Carefully break the eggs into a small bowl.
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17
Slide the eggs, 1 at a time, into the water and poach just until the white is set and the center is still soft, about 3 minutes.
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18
Remove the poached eggs from the water with a slotted spoon and place on a paper towel lined plate.
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19
Remove the brioches from the oven and place 1 on each of 4 plates.
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20
Spoon the chicken hash into each brioche, sprinkle with cheese and top with poached eggs.
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21
Set the brioche tops on each at an angle, and sprinkle with the parsley.
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22
Serve.
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23
Place the potato cubes in a medium, heavy saucepan and add enough water to cover.
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24
Bring to a boil and cook until slightly tender, about 10 minutes.
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25
Drain and pat dry with paper towels.
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26
Heat an electric fryer to 360 degrees F. Add the potatoes and, fry, turning to brown evenly, for 3 to 4 minutes.
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27
Drain
