Heat oil in a 12-inch skillet over medium heat.
Season chicken with ¼ teaspoon salt and ¼ teaspoon pepper. Add chicken to skillet. Cook until chicken is no longer pink in the center and lightly browned, turning once halfway through, 6 minutes. Remove chicken from skillet; keep warm.
Add mushrooms to skillet. Cook until mushrooms are browned and their juices have evaporated, about 4 minutes.
Stir in garlic. Cook and stir until fragrant, 1 minute.
Add broth, vinegar, honey, mustard, and thyme to the skillet. Bring to a boil, stirring to scrape up browned bits with a wooden spoon. Reduce heat and simmer until liquid is reduced by half, about 4 minutes.
Whisk butter into sauce until incorporated. Season with remaining ¼ teaspoon salt and ¼ teaspoon pepper.
Return chicken to skillet, cover and heat through until a thermometer inserted into the thickest portion registers 165 degrees F (74 degrees C), 3 to 4 minutes.
Garnish with additional fresh thyme.
