Preheat a convection oven to 200°C (400°F).
Heat 1 tablespoon of olive oil in a skillet over medium-high heat. Add minced mushrooms and sauté until all moisture has evaporated and mushrooms are browned.
Add shallots and garlic to the mushrooms; sauté for 2 minutes until translucent. Remove from heat and stir in thyme, cream cheese, and Parmigiano-Reggiano. Season with half the salt and pepper.
Using a sharp boning knife, cut a horizontal pocket into the thickest part of each chicken breast, ensuring the blade does not penetrate the opposite side or the edges.
Equally distribute the mushroom mixture into the pockets of the four chicken breasts. Secure the openings with wooden toothpicks.
Season the exterior of the chicken breasts with the remaining salt and pepper.
In an oven-safe skillet, heat the remaining oil and butter over medium-high heat. Sear the chicken for 3 minutes per side until a golden-brown crust forms.
Transfer the skillet to the preheated oven and roast for 12 to 15 minutes, or until the internal temperature reaches 74°C (165°F).
Remove from the oven and allow the meat to rest for 5 minutes before removing toothpicks and serving.