Heat oil in a large skillet over med-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 2-3 min 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 minute.
Dissolve cornstarch in remaining 1/2 cup of broth in small bowl, add to skillet. Simmer until sauce thickens, stirring constantly and incorporating mushrooms into the liquid, about 1 minute.