Preheat the oven to 350°F (175°C).
Pat the chicken thighs dry and season generously on both sides with salt and black pepper.
Heat the olive oil in a large oven-safe skillet over medium heat. Place the chicken skin-side down and cook for about 3–4 minutes per side, or until nicely browned. Transfer the chicken to a plate.
Add the butter to the same skillet. Add the sliced shallots, season lightly with salt and pepper, and cook for 3–4 minutes, stirring often, until softened.
Add the garlic, fresh thyme, and red chili flakes. Cook for about 30 seconds, stirring constantly, until fragrant.
Pour in the balsamic vinegar and 1/2 cup of the chicken stock. Stir in the Dijon mustard and honey. Let the sauce simmer for about 1 minute.
Return the browned chicken thighs to the skillet, skin-side up. Add the halved figs around the chicken.
Transfer the skillet to the oven and bake for 15–20 minutes, or until the chicken is cooked through and reaches a safe internal temperature of 165°F.
If desired, broil for the final 1–2 minutes to deepen the color of the chicken skin. Watch carefully so the figs and sauce do not burn.
Remove the skillet from the oven carefully. If the sauce is too thick, stir in a little of the remaining chicken stock. Finish the Skillet Balsamic Chicken with fresh lemon juice and serve immediately.