Loved it and haven't made it in years.
Mine was a little different -- chicken thighs pounded or cut horizontal to make them thinner. More flavor and breast is too dry for me. Pan fried in olive oil only and this was the dish that taught me how to not burn garlic. A bunch of recipes for various things will tell you to chop garlic and put it in at the beginning of cooking a protein to golden brown. By the time a protein is golden brown, minced garlic will burn so I adopted the other technique which is faster too -- smash the cloves unpeeled and then pop them out of the skins and add to the olive oil as it is heating. Flavor the oil and either pull them out before the protein goes in or grab them before they turn. Set them aside, chop and return them to the sauce at the end.
I have never tried it with thyme but I should. Usually use flat leaf parsley if I have any.
Mushrooms were the other tricky part of this one. Had to find the balance between browning them versus not drying them out. If the wine goes in before they color then they won't color and they just kind of poach. I would do the mushrooms in butter and the chicken in olive oil.
If you make angel hair pasta at the same time them you can finish that pasta in the sauce. So that is three pans simultaneously -- pasta water, chicken and mushrooms/sauce. After you pull the chicken you can quickly cook julienne-cut carrots or spinach in that pan.
Side note: during Prohibition marsala wine got a pass because it was marketed as medicine.
Did I mention I have always been obsessed with this dish?