Wipe the chicken thighs with paper towels. You want them dry so they'll brown better. Rub with salt and pepper.
Heat a non-stick frying pan. Add the chicken thighs in a single layer, skin side down. Cook over high heat until the skins are nicely browned. Flip to brown the opposite side. Scoop out the chicken thighs and transfer to a plate. Keep warm.
Melt the butter in the frying pan. Over medium heat, saute the garlic until just starting to brown (the smell will whet your appetite). Add the mushrooms and cook for about a minute.
Return the chicken to the pan. Add the tarragon. Pour in the lemon juice and cream. Add the lemon zest. Stir. Taste the sauce and add more salt and pepper, as needed. Set the stove to the lowest setting. Cover the pan and braise the chicken thighs for 10 to 15 minutes. They are partially cooked already so it won't take all that long for them to get cooked through.
The cream would have curdled at this point and the dish may not look too pretty. Don't worry. That's what the milk is for. Pour in the milk and stir. Start with half a cup. The sauce should start to look silky. If it's still too thick for you, add another quarter cup of milk. Once the sauce is bubbly, turn off the heat. Taste once more and adjust the seasonings. Stir in the chopped parsley.
Serve the creamy lemon garlic chicken over rice or pasta.