Preheat oven to 375°F and line a baking sheet with foil. Spray lightly with nonstick cooking spray.
Mix the filling by combining softened goat cheese, chopped pistachios, dried figs, minced garlic, thyme, onion powder, salt, and pepper in a small bowl until well incorporated.
Cut pockets in chicken by slicing horizontally into the thickest side of each breast, about 3/4 of the way through. Season outside with salt and pepper.
Stuff the chicken by dividing the goat cheese mixture evenly among the four breasts. Secure openings with toothpicks if needed.
Dredge in flour by coating both sides of each stuffed chicken breast lightly, tapping off excess.
Sear the chicken in 2 tablespoons olive oil over medium-high heat for 3-4 minutes per side until golden brown.
Bake the seared chicken on the prepared baking sheet for 10-12 minutes until internal temperature reaches 165°F.
Make the sauce by adding chicken broth, fig jam, balsamic vinegar, and rosemary to the skillet. Simmer 5 minutes until thick and syrupy. Stir in butter until smooth.
Serve the stuffed chicken with the fig balsamic sauce drizzled over top. Garnish with fresh herbs if desired.