Pat the chicken dry and season both sides with salt and black pepper. Lightly coat each chicken breast with flour, shaking off any excess.
Heat the olive oil in a large oven-safe skillet over medium-high heat. Add the chicken and sear for 4 to 5 minutes per side, until deeply golden. Transfer the chicken to a plate; it will finish cooking in the sauce.
Reduce the heat to medium and add the butter to the same skillet. Add the mushrooms and cook undisturbed for 3 to 4 minutes before stirring. Continue cooking until the mushrooms are deeply browned and their moisture has cooked away.
Add the shallot, garlic, thyme, sage, and red pepper flakes. Cook for 2 to 3 minutes, stirring frequently, until the shallot is tender and the garlic is fragrant.
Pour in the white wine and scrape up the browned bits from the bottom of the skillet. Let it simmer for 2 to 3 minutes, allowing the wine to reduce slightly.
Stir in the chicken broth and heavy cream. Bring the sauce to a gentle simmer, then add the Brie and Parmesan. Stir until the cheeses melt completely and the sauce becomes smooth and creamy.
Return the chicken and any accumulated juices to the skillet. Spoon some of the mushroom sauce over the chicken and simmer gently for 8 to 10 minutes, or until the chicken reaches 165°F (74°C) internally.
Remove from the heat and let the chicken rest for a few minutes. Sprinkle with fresh parsley and additional thyme before serving.