How to Make It
Preheat the oven to 375°. In an ovenproof skillet, melt the butter in 2 tablespoons of the vegetable oil. Add the potatoes, cut side down, season with salt and pepper and cook over high heat until browned, 3 minutes. Turn the potatoes and roast in the oven for 20 minutes. Add the mushrooms, season with salt and pepper and roast, stirring a few times, for 25 minutes. Stir in the rosemary, thyme and oregano; keep warm.
Meanwhile, in another ovenproof skillet, heat 2 tablespoons of the vegetable oil. Season the chicken with salt and pepper and add to the skillet, skin side down. Cook over high heat until browned, 3 minutes. Turn the chicken and transfer to the oven. Roast for 10 minutes, until just cooked through. Transfer the chicken to a carving board; let rest.
In a nonreactive saucepan, combine the wine, vinegar and shallot and boil until reduced to 1 tablespoon, 2 minutes; scrape into a blender. Add the egg yolks and blend. Add 1/2 teaspoon of the lemon juice and the mustard and blend. With the machine on, slowly pour in the remaining 1/2 cup of vegetable oil; blend until smooth. Season the dressing with salt and pepper; stir in the tarragon.
In a bowl, combine the olive oil with the remaining 3 teaspoons of lemon juice and season with salt and pepper. Add the watercress, arugula, cucumber and onion and toss. Add the warm potatoes and mushrooms and toss. Slice each chicken breast crosswise and serve with the salad, passing the béarnaise dressing at the table.