Preheat oven to 375 degrees then line a half sheet pan with nonstick sprayed foil and set aside.
In a small bowl, mix together goat cheese, dried figs, pistachios, garlic and onion powders, salt, and pepper. Using a sharp knife, cut a pocket horizontally into each chicken breast then stuff with 1/4 of the chèvre and fig mixture. Close opening with toothpicks if desired.
Combine flour, salt, and pepper in a shallow dish then dredge both sides of the chicken breasts, tapping excess flour off.
Add extra virgin olive oil to a large skillet over medium-high heat. Add chicken then saute until golden brown on one side, 3-4 minutes, and then flip and saute until golden brown on the other side, 2-3 minutes. Transfer chicken to sheet pan then bake until cooked through, 10-12 minutes.
Meanwhile, add chicken broth, fig jam, balsamic vinegar, and dried rosemary to skillet then turn heat down to medium and simmer until sauce is thick and syrupy, 5 minutes, stirring occasionally. Add butter then stir until smooth, and then taste and add salt and pepper if necessary. Plate chicken then drizzle with sauce and serve.
Notes
Feel free to stuff the chicken up to 2 days ahead of time.