Pat chicken thighs dry and season both sides with salt and pepper.

Heat olive oil in a large skillet over medium-high heat until shimmering. Add chicken thighs and cook, 3 to 4 minutes on each side. Add figs, chicken broth, balsamic vinegar, and shallot. Bring contents to a boil, cover, reduce heat, and let simmer until chicken thighs are no longer pink in the centers and an instant-read thermometer inserted into the centers reads at least 165 degrees F (74 degrees C), about 8 minutes. Transfer thighs to a serving plate and keep warm.

Remove figs from the skillet using a slotted spoon and place them around the chicken on the plate.

Add minced rosemary to the skillet contents, increase heat to medium-high, and allow contents to come to a full boil to create a the reduction. Continue to boil, stirring often, until liquid reduces to approximately 1 cup, 8 to 10 minutes. Drizzle balsamic reduction over chicken and figs and garnish with rosemary sprigs. Serve warm.