Combine the flour, salt, and pepper in a shallow bowl.
Heat the olive oil in a very large skillet over medium-high heat. Dredge the chicken cutlets in the flour mixture. Once the oil is hot, cook the cutlets in the pan for about 3 minutes on each side until they are golden brown and almost but not quite cooked through (a bit pink in the center). Transfer the cutlets to a large plate as they are cooked.
Turn the heat down to medium and add the mushrooms, onions, and garlic. Cook, stirring frequently, for about 7 minutes, until any liquid that the mushrooms have released has evaporated and they are tender and lightly browned. Add the chicken broth and stir so that any browned bits stuck to the bottom of the pan are released into the liquid. Stir in the cream, thyme and Parmesan. Bring to a simmer and cook, stirring frequently, until the sauce thickens, about 4 minutes.
Tuck the chicken back in the pan, with any juices that may have collected on the plate, making sure the chicken is at least mostly covered by the sauce. Cook for 2 more minutes until the chicken is hot and cooked through.
Either serve the chicken over the hot pasta or rice or transfer it to its own serving dish. Sprinkle with the parsley, and serve hot with extra Parmesan passed on the side.
Notes
Morels are delicate, they are expensive, and they have such a haunting smoky, nutty, earthy flavor that mucking around with them too much would be a shame. They can be cultivated but most often found in the wild, and they are also available dried.Dried morels are awesome because they are available year-round. They store for months and months. If you don’t have morels or don’t feel like making this a pricy dish, just stick with any old sliced mushrooms, even button or cremini. I did a 50/50 morel/cremini thing because ½ pound is what I had on hand.