Slice chicken breasts into smaller "cutlets" or pound out thicker chicken breasts so that they will cook more quickly. Rinse and pat dry with paper towels.
In a large skillet, heat 2 tablespoons of butter and 2 tablespoons olive oil over medium-high heat until sizzling.
Mix together flour, salt, and pepper in a shallow bowl. Dredge each piece of chicken on both sides with flour mixture and set aside.
Place floured chicken into the skillet in a single layer. Brown on each side (about 5 minutes per side). Cook the chicken in batches to avoid overcrowding in the pan. When chicken has browned on each side, remove to a plate/bowl and keep warm.
Add two remaining tablespoons of butter to the skillet. When melted, add the mushrooms and onions. Cook without disturbing for 3-4 minutes or until mushrooms are dark on one side.
Stir mushrooms and continue cooking over medium heat until any liquid has been absorbed/evaporated and the mushrooms are browned on both sides.
Add minced garlic to the mushrooms and saute 1-2 minutes longer until garlic begins to brown, but not burn.
Pour wine into pan and stir to scrape all of the brown bits off the bottom of the pan to deglaze. Simmer 1-2 minutes.
Add chicken broth and thyme sprigs to the pan and bring to a boil. Simmer over medium heat for about 5 minutes.
Whisk cornstarch into a few tablespoons of water until dissolved. While stirring the sauce in the skillet, drizzle the cornstarch "slurry" into the sauce and continue stirring to distribute evenly.
Return chicken to the skillet and simmer for 3-4 minutes longer, until chicken has warmed through and sauce thickens.
Garnish with additional fresh thyme, salt, and freshly ground pepper.
Serve with mashed potatoes, over noodles or rice, or with parmesan french bread to soak up all that yummy sauce.