Pat chicken dry, season with salt and pepper. Heat oil in a large skillet over medium-high heat.
Brown chicken skin-side down 6–8 minutes until golden; flip and brown 3 minutes. Transfer to a plate.
Add shallots to skillet, cook 2–3 minutes until soft. Stir in figs, chicken stock, balsamic, honey, and Dijon.
Return chicken to skillet, skin-side up. Simmer, covered, 18–22 minutes until internal temp reaches 165°F.
Remove chicken; stir butter into sauce and reduce 2–3 minutes until slightly thickened. Spoon sauce over chicken and garnish with thyme.