Preheat oven to 350-degrees Fahrenheit (175 degrees Celsius).
To brine the chicken:
Pat the chicken dry with paper towels and place in a large bowl.
Add the water and 2 tablespoons kosher salt to the bowl with the chicken and allow to sit for at least 30 minutes, but no longer than 90 minutes.
Once brined, pat the chicken dry with paper towels, season each with black pepper to taste, and discard the brining water.
To prepare the mushrooms:
Heat 2 tablespoons of the oil in the skillet over medium heat. Add the mushrooms and stir well. Cook for 2-3 minutes before adding the shallots and 1 tablespoon of the chopped tarragon.
Cook the mushrooms, shallots, and tarragon together for an additional 3-4 minutes, stirring occasionally before adding ½ teaspoon salt and ¼ teaspoon black pepper. Stir again and remove the sauteed mushrooms from the skillet using a slotted spatula.
To brown the chicken:
In the same skillet, add the remaining 1 tablespoon of avocado oil to the skillet along with the dried, brined chicken breast. Allow the chicken breast to brown for 3-4 minutes before flipping it over to brown on the second side for an additional 2-3 minutes.
To complete the dish:
Once the chicken is browned on both sides, remove it to a plate. Add the white wine to the skillet and scrape up the browned bits (fond) from the bottom of the pan to deglaze. Stir in the heavy cream, chicken stock, butter, remaining tarragon and prepared mushrooms to the pan, and cook for 1-2 minutes on medium/high heat before adding the chicken into the pan.
Spoon some of the sauce over the chicken and place the oven safe skillet into the oven (uncovered) for 15-20 minutes, or until the chicken reaches on internal temperature of 165-degrees Fahrenheit on an instant read thermometer. Allow the chicken to sit in the skillet for 5-7 minutes.
Remove the chicken breast from the pan and spoon the sauce and mushrooms over the chicken before serving.
Notes
Store any leftovers in an airtight container in the refrigerator for up to 5 days.May be frozen after cooking. Cool completely and store in an airtight freezer container or bag for up to 3 months.