In a 10 to 12-inch cast-iron skillet, heat the oil and melt the butter, over medium to high heat. Flatten the chicken slightly, by placing it between layers of plastic wrap or placing into a plastic bag, making it even thickness. Dry the chicken breasts well, using paper towels. Otherwise, it will not brown properly.
Brown the chicken breasts in the hot oil and butter. Cook on each side about 4-5 minutes. Use tongs to turn the chicken. Place the cooked breasts on a plate, season with lemon pepper seasoning and set aside.
Prepare the pasta, by bringing a pot of water to a boil, season with salt. Add the pasta and boil for about 10-12 minutes, until al dente. Drain the pasta before adding it to your cream sauce.
While the pasta is cooking, brown the mushrooms and shallots in the pan you cooked the chicken. Over medium to high heat. Use a wooden spoon to stir the mushrooms occasionally.
Once they are browned, deglaze the pan with the wine. Bring to a boil. Then add the cream.
Stir occasionally with a wooden spoon. Let cook on high heat for about 8 to 10 minutes or until the sauce has thickened.
Remove from the heat. Add the lemon juice. Season with salt and pepper as needed. Cut the chicken into strips and place back in the cream sauce. Add the pasta and stir. Add the chopped rosemary and stir. Enjoy immediately.