Stir together olive oil, lemon juice, garlic, rosemary, and salt and pepper in a large bowl. Cut each chicken breast into 2 equal pieces, the slice horizontally. With a meat mallet, gently pound the chicken into even cutlets, using the flat side of mallet. This works best if you start in the middle and pound out to the edges. Or purchase 4 chicken cutlets. Place the chicken in the bowl. Stir together to coat well and refrigerate for 15 to 30 minutes.
Remove the chicken from the marinade and place on a plate. Discard marinade. Season the chicken with salt and pepper on one side only.
On a plate combine place the flour. Dredge lightly in the flour (you will not use all of it) and tap the breasts to remove excess.
Heat a wide non-stick fry pan over medium high heat. Then add oil. When oil is hot, place one or two pieces of chicken in the pan – however many will fit without crowding. Cook for 2 minutes, until bottom is browned in spots. Turn over and brown other side, about 1 1/2 minutes. (Do not overcook or the chicken will be dry.)
Transfer to a platter or sheet pan and keep warm in a low heat oven, or simply tent with foil. If there is more than a tablespoon of fat in the pan, pour some (but not all) it off into a jar or bowl.
Turn burner heat down to medium. Add mushrooms to the pan. Let them sear for about 30 seconds to a minute without moving them, then stir, scraping the bottom of the pan with a wooden spoon to deglaze. When mushrooms have softened slightly and begun to sweat, add wine and thyme and salt and pepper to taste. Continue to stir until wine has reduced some and mushrooms are tender and browning, 5 to 10 minutes. Spoon over the chicken, and serve.