Directions

1. Preheat an oven to 350 degrees F (175 degrees C).
2. Melt 1/2 cup butter in a large skillet; cook and stir the onion in the butter until caramelized, about 10 minutes. Stir in the sugar and cook until the sugar completely melts, about 5 minutes. Remove from heat and set aside to cool completely.
3. Mix together the flour, salt, and pepper in a shallow dish. Whisk together the egg and water in a small bowl. Dredge each chicken breast in the flour; shake off the excess flour and then dip the chicken in the egg and water mixture.
4. Heat the olive oil in a large skillet over medium-high heat; cook the chicken in the hot oil until completely browned, 5 to 7 minutes per side.
5. Place 2 of the chicken breast halves in the bottom of a baking dish. Top each with about half the brie and half the caramelized onions. Lie the remaining chicken breast halves atop the brie and onions.
6. Bake the chicken in the preheated oven until the cheese is melted and the chicken is no longer pink in the center, about 10 minutes.
7. While the chicken bakes, melt 2 tablespoons butter in a small skillet over medium heat. Allow the butter to brown slightly before adding the vermouth; bring to a simmer and allow to cook until the liquid reduces by about half. Season with the thyme, salt, and pepper. Stir the arrowroot into the mixture to thicken. Serve the sauce over the chicken.