Soak figs in hot water for 5 minutes. Meanwhile, combine walnuts and thyme in a small bowl. Drain figs and add to bowl with walnuts; stir.
Melt the butter in a large nonstick skillet over medium-high heat. Season one side of the chicken cutlets with 1/8 teaspoon each of the salt and pepper.
Place chicken, seasoned-side down, in skillet and cook 3 to 4 minutes. Season top side with remaining 1/8 teaspoon each salt and pepper. Flip over chicken and cook an additional 3 to 4 minutes. Remove to a serving plate and keep warm.
Add onion to skillet and cook 2 minutes. Add fig mixture and chicken broth to skillet and cook 1 minute. Remove skillet from heat, then spoon fig compote over chicken. Serve with roasted potatoes and salad, if desired.