To make the stuffing, in a small sauté pan over medium heat, warm the olive oil. Add the shallot and sauté until it begins to soften, 1 to 2 minutes. Add the mushrooms and continue to sauté, stirring occasionally, until the mushrooms are tender, about 4 minutes. Let cool.
Meanwhile, in a bowl, stir together the cheese, parsley, chives, tarragon and a pinch of nutmeg. Stir in the cooled mushroom mixture, and season to taste with salt and pepper.
Preheat an oven to 220°C.
Pat the chicken breasts dry with paper towels. Carefully slide your fingers under the skin on each breast, separating it from the meat but leaving it attached on one side. Spoon about 2 Tbs. of the stuffing directly onto the meat, spreading it evenly, and pull the skin back in place to cover the filling. Flatten the filling evenly by gently pressing on the skin. Arrange the stuffed breasts skin side up in a roasting pan large enough to hold them in a single layer. Brush with 1 tsp. oil and sprinkle with salt and pepper.
Roast for 15 minutes. Reduce the oven temperature to 190°C. Continue to roast, basting every 10 minutes with the pan juices, until the skin is browned and an instant-read thermometer inserted into the thickest part of a breast registers 77°C, about 30 minutes.
Transfer the chicken to a plate and keep warm. Place the roasting pan on the stove top over high heat. Add the wine and deglaze the pan, stirring to remove any browned bits from the bottom of the pan. Bring to a boil and cook until reduced by half, about 5 minutes. Using a spoon, skim and discard the fat from the pan juices, then strain the juices through a fine-mesh sieve into a warmed small pitcher.
Arrange each chicken breast on a warmed individual plate and serve immediately, passing the pan juices at the table. Serves 6.