You're going to need a deep frying pan, plus olive oil and a lump of butter.
Pour the flour, salt and black pepper in a bowl, stir. Dip the chicken breasts in it and flour them well on both sides. Don't shake them off, leave aside.
Heat the olive oil on medium heat and braise the chicken breasts on both sides to a nice golden color. Take them out.
Pour the mushrooms in the pan and the Marsala wine over them immediately. Leave them to saute about 4-5 min.
Return the chicken breasts to the pan and start flipping them over constantly with tongs - about 15 min. The sauce in the pan will turn dark and highly fragrant because of the Marsala.
Serve the chicken breasts in a suitable plate and pour the sauce over them.
Serve with mashed potatoes as garnish.
Optional: Add another lump of butter to the pan, plus 1/2 tsp dry oregano and 1/4 cup dry sherry. Pour in about 1/2 cup cooking cream. Leave to boil for a few min. and pour it over the chicken breasts.
Source: Video of an original Italian recipe from Marsala, Sicily.
Notes: When adding the wine you may get a flambe effect.
You can halve the chicken breasts and tenderize them lightly in plastic wrap but be careful so they don't tear. I did not do this because thicker pieces of chicken retain more juice and moisture, which is important because white chicken meat is quite dry.