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1
Preheat oven to 400 degrees F.
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2
Season the chicken pieces on both sides with 1/2 teaspoon of the salt and 1/4 teaspoon of the pepper.
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3
Heat the oil in a large ovenproof skillet or saute pan over medium-high heat.
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4
Add the chicken pieces and cook until golden brown on both sides, 4 minutes on the first side and 3 minutes on the second side.
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5
Transfer the pan to the oven and roast until the chicken is cooked through, 18 to 20 minutes.
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6
Meanwhile, melt the butter over medium-high heat.
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7
Add the garlic and cook, stirring until fragrant, about 30 seconds.
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8
Add the ham, mushrooms, green onions, and the remaining 1/4 teaspoon salt and 1/8 teaspoon pepper and cook, stirring, until the mushrooms give off their liquid and start to turn golden brown, 3 to 4 minutes.
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9
Add the wine and bring to a boil, stirring to deglaze the bottom of the pan.
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10
Reduce the heat to medium and simmer until the liquid is reduced by two-thirds in volume, 4 to 5 minutes.
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11
Add the Brabant potatoes and parsley and cook, tossing gently to mix, until the potatoes are warmed through, 1 to 2 minutes.
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12
To serve, place equal amounts of the vegetable mixture in the center of four serving plates and top with the chicken.
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13
Spoon the Bearnaise sauce over the chicken and serve immediately.
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14
Combine the shallots, tarragon sprigs, white wine, and vermouth in a small saucepan, bring to a boil, and cook until reduced to 4 teaspoons.
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15
Strain through a fine-mesh sieve and let cool.
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16
In the top of a double boiler, or in a metal bowl fitted over a pot of simmering water, whisk the egg yolks and reduced wine mixture until ribbons start to form when the whisk is lifted.
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17
Whisking constantly, drizzle in the melted butter, a bit at a time, until the mixture thickens.
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18
Remove from the heat, add the lemon juice, chopped tarragon, salt, and pepper and whisk to blend.
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19
Adjust the seasoning, to taste.
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20
Serve immediately or keep warm, covered, over a pot of simmering water, for 5 to 10 minutes.
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21
Whisk before serving.