This is a simplified take on a French dish, poulet a l'estragon, or tarragon chicken, and it can be more, or less over the top than the original, depending on how hard you want to push it. For example, Nigella Lawson's version calls for a third of a cup of vermouth; I used more to get a more concentrated reduction. I did use her dried tarragon, but won't next time. All it added was a sort of pushback from the sauce that didn't work. Stick with fresh tarragon, it is as flavorful as the dried but better behaved.
As per usual, this is my take on a bunch of similar recipes culled from the internets. I'm no chef, just an old line cook. Use the salt and pepper to season every layer. You know to do this.
2 chicken breasts
a bottle of good dry vermouth
a big shallot
2 cups of heavy cream
This is so simple, it isn't even really French cooking, much more like Italian. It's completely ingredients driven, so buy the best.
Mince the shallot.
Mince about 1/4 cup of the tarragon leaves. Keep some unminced leaves aside. You might mince the tarragon leaves at the last moment to keep them as herbacious as possible before you saute them. I do this.
Saute the chicken breasts, set aside covered in foil to keep them warm Undercook them a bit so they don't toughen up as they rest, but don't poison anyone with raw chicken.
Add the minced shallots and tarragon to the pan. Saute them for about 1 minute.
Leave the shallots and tarragon in the pan. Deglaze the pan with about 1 cup of vermouth.
Reduce the vermouth by about 80%.
Add two cups of heavy cream to the pan. Reduce to about 1/2 cup, or thickened enough to coat a spoon. If you reduce too far, add some cream. Take the sauce off the heat.
Rough chop the remaining tarragon leaves.
Plate the chicken, and peas. Add a small squeeze of lemon juice to the sauce to brighten it,and stir it in. Sauce the chicken. Sprinkle some of the fresh chopped tarragon over the chicken. Serve with peas and pearl onions, or asparagus, or whatever seasonal vegetable you like.