Heat oil in a large skillet over medium-high heat. Add mushrooms and sauté until tender and releasing liquid, about 5 minutes.
Meanwhile, place chicken on a plate and season both sides with thyme, salt and pepper. Move mushrooms to outer edge of skillet once cooked. Place chicken in center of skillet and sauté until golden, about 2 to 3 minutes per side.
Add wine to skillet; simmer 1 minute. Add 3/4 cup of broth and simmer, uncovered, until chicken is tender and cooked through, about 8 minutes.
Dissolve cornstarch in remaining 1/2 cup of broth in a small bowl; add to skillet. Simmer until sauce thickens, stirring constantly and incorporating mushrooms into the liquid, about 1 minute. Serve chicken with mushroom sauce spooned over top. Yields about 3 ounces of chicken and 1/3 cup of sauce per serving.