In a large nonstick frying pan on medium-high heat, melt 1-1/2 tablespoons butter. Add mushrooms, lemon zest, and lemon juice. Cover and simmer 5 minutes to allow juices to accumulate.
Uncover and cook over high heat, stirring often, until juice evaporates and mushrooms brown, about 5 minutes. Add broth and stir to release browned bits to pan. Pour into a small bowl, scraping pan well.
Trim fat from thighs. Working with one thigh at a time, place on a piece of plastic wrap. Cover with another piece. Pound meat with a flat-surface mallet (you can use a rolling pin if you don’t have a mallet) until about 1/4 inch thick. Pull of plastic and season with salt and pepper. Repeat with remaining thighs.
Add 1 tablespoon butter to pan and melt over high heat. Lay chicken pieces in pan (you may have to cook in batches). Cook until edges begin to turn white and bottom is lightly browned, 2 to 4 minutes. turn over and cook until meat is no longer pink in the center, 6 to 8 minutes total (cut to test), lifting to a platter as cooked. Cut into serving portions and cover with foil to keep warm.
Pour mushrooms and broth into pan and add 1/4 cup parsley. Stir in wine, scraping to release brown bits. Immediately add remaining butter and stir rapidly until melted. Drain juice from cooked chicken into pan. Bring to a boil, stirring. Reduce the broth slightly. Add vanilla and blend well. Adjust seasonings if necessary. Pour over chicken and sprinkle with remaining parsley.
*If you are not using a non-stick frying pan, add a tablespoon of olive oil or other high quality oil to the butter when you first heat the pan. You will need to add additional oil when you cook the chicken.