Preheat oven to 400 degrees F.
Cut the chicken into 6 pieces, 2 breasts, 2 thighs, and 2 legs, and remove the bones from the breasts. Season the chicken pieces on both sides with 1/2 teaspoon of the salt and 1/4 teaspoon of the pepper.
Heat 1 tablespoon of the oil in a large ovenproof skillet or saute pan over medium-high heat. Add the chicken pieces and cook until golden brown, 4 minutes on the first side and 3 minutes on the second side. Place 2 tablespoons of butter on the chicken and transfer the pan to the oven and roast until the chicken is cooked through, 18 to 20 minutes.
Meanwhile, heat the remaining 1 tablespoon of olive oil over medium-high heat. Add the garlic and cook, stirring, until fragrant, about 30 seconds. 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. Add the wine and bring to a boil, stirring to deglaze the bottom of the pan. Reduce the heat to medium and simmer until the mixture is thickened, 4 to 5 minutes. Add the Brabant Potatoes and parsley, tossing gently to mix, and cook until the potatoes are warmed through, 1 to 2 minutes.
To serve, place equal amounts of the vegetable mixture in the center of 4 large plates and top with the chicken. Spoon the Bearnaise Sauce over the chicken and serve immediately.
Combine the white wine, vermouth, red wine vinegar, shallots, tarragon sprigs, and peppercorns in a small saucepan, bring to a boil, and cook until reduced to 4 teaspoons. Strain through a fine-mesh sieve and let cool.
Whisk the egg yolks in the top of a double boiler (or in a medium bowl set over a pot of barely simmering water) until thick and pale yellow. Remove the double boiler (or bowl and saucepan) from the heat and gradually add the butter, whisking constantly to thicken. Add enough tepid water to thin to pouring consistency. Add 4 teaspoons of the wine reduction, the chopped tarragon, the salt, and hot sauce, and whisk well to blend. Adjust seasoning to taste.
Serve immediately, or cover and keep warm for up to 10 minutes, whisking occasionally to keep from separating.
Recipe courtesy of Emeril Lagasse, originally appearing in Emeril's Delmonico cookbook, William Morrow Publishers, New York, 2005