If your chicken breasts have tenderloins attached, remove them and reserve for another use. Place the breasts between 2 sheets of plastic wrap and gently pound them to a uniform thickness of 1/2-inch. Season on both sides with salt and pepper.
Spread the flour on a sheet of wax paper, lightly dredge the chicken breasts and shake off any excess flour.
Heat the oil in a large pan over medium-high heat. Add the chicken in a single layer and cook until pale golden, 1-1/2 to 2 minutes per side. Transfer to a plate and set aside.
Add the butter to the pan and reduce the heat to medium. Add the shallots and sauté until soft and fragrant, 2 to 3 minutes. Stir in the chicken broth, orange zest and rosemary. Bring the mixture to a simmer, add the figs and orange segments, season lightly with salt and pepper and cook until the figs are tender, 2 to 3 minutes.
Return the chicken to the pan and, using tongs, turn the pieces several times to coat them with the sauce. Continue cooking until the sauce thickens slightly and the chicken is cooked through, 2 to 4 minutes longer.
Arrange the chicken on a serving platter and spoon the sauce over top. Serve with mashed potatoes, rice or buttered orzo.