Using a meat tenderizer, pound the chicken breasts to an even thickness.
Salt and pepper both sites of each chicken breast.
Place a sauté pan with one tablespoon of the olive oil over medium heat. Make sure the bottom of the whole pan is coated with the olive oil. Add the chicken breasts. Cook for 1 minute until golden on the bottom. Turn the breasts over and reduce heat to low.
Cover the pan with a tight-fitting lid. Cook for 10 minutes (check time). Do not remove the lid while cooking. After 10 minutes, remove the pan from the heat, still covered, and allow to sit for 10 additional minutes.
Make sure there is no pink in the middle of the breasts or that a meat thermometer reads 165 degrees F when inserted in the center. Remove chicken from the pan and set aside, covered to stay warm. Add one tablespoon olive oil and the mushrooms and onions to the pan and increase heat to medium.
Allow mushrooms and onions to cook 5 to 8 minutes until the water releases from the mushrooms and evaporates. Whisk in two tablespoons flour and allow to cook for about 1 to 2 minutes, until fragrant.
Add in the garlic and cook for an additional 30 seconds. Add a 1/4 cup of milk, thyme, salt, and pepper. Whisk until thickened. Add another 1/4 of a cup and whisk until thickened. Repeat until all of the sauce has thickened.
Place chicken on serving plate. Pour sauce over the chicken and serve.