In a shallow bowl, combine the flour and 1/4 teaspoon each salt and pepper. Coat the chicken, shaking off any excess.
In a large, heavy skillet, heat the olive oil over medium-high heat. Working in batches, add the chicken and cook, turning once, until just cooked through, about 5 minutes; transfer to a plate. Add 1/2 tablespoon butter and the mushrooms to the pan and cook, stirring, until browned, about 5 minutes; transfer to the plate of chicken. Stir in the shallot and the remaining 1 tablespoon butter. Add the wine and chicken broth and simmer until reduced by one-third, about 5 minutes.
Return the chicken and mushrooms to the pan, add the thyme and cook, turning, to heat through; season with salt.