Heat the extra virgin olive oil in a large skillet over medium heat. Add the diced onion and minced garlic, stirring occasionally until they become translucent and fragrant, about 3-4 minutes. Add the fresh spinach in batches, stirring until it wilts completely. Stir in the rehydrated cranberries, dried thyme, balsamic vinegar, salt, and black pepper, cooking for just 1 minute to combine the flavors. Transfer the mixture to a bowl and let it cool slightly—this makes it easier to work with when stuffing. I like to rehydrate my cranberries in warm water for about 5 minutes before adding them; it plumps them up and makes them juicier in the filling.
Lay each chicken breast flat on a cutting board. Working with one breast at a time, place a slice of brie (rind already removed) inside the pocket, then add about 2 tablespoons of the spinach filling from Step 1 on top of the brie. Fold the chicken closed and secure tightly with kitchen twine to prevent the filling from leaking during cooking. Season the outside of each stuffed chicken breast evenly with the salt and black pepper, making sure to season all sides.
Preheat your oven to 375°F. While it heats, warm the olive oil in a large oven-safe skillet over medium-high heat until it shimmers. Working in batches if needed to avoid overcrowding, carefully place the stuffed chicken breasts in the hot skillet and sear for 3 minutes on each side until golden brown. This browning develops flavor and creates a protective crust that helps keep the chicken juicy. Don't move the chicken around too much—let it sit undisturbed so it develops that beautiful golden color.
Transfer the skillet with the seared chicken to the preheated 375°F oven and bake uncovered for 25 minutes, until the chicken reaches an internal temperature of 165°F when measured with a meat thermometer at the thickest part. Remove from the oven and let the chicken rest in the skillet for 5 minutes—this allows the juices to redistribute throughout the meat, keeping it moist and tender. After resting, carefully remove the twine from each breast and serve immediately with the pan juices drizzled over the top.