Heat two large skillets over medium high heat. Add enough olive oil to each pan to coat the bottom (between 1 and 1 ½ tablespoons).
Liberally season chicken breasts with salt and pepper. Add to one skillet and cook for about 5 minutes on one side. Turn the heat down to medium, flip the chicken breasts over and cook the other side until browned on the outside and cooked through about 3-4 minutes. Remove to a plate to rest a few minutes while finishing the sauce.
In the other pan add the chopped onions. Turn heat down to medium low and cook until onions are soft and starting to caramelize (about 5 minutes). Add the chopped garlic and cook for 30 seconds. Add the mushrooms and cook for about 5 minutes until browned. Season with a good pinch of salt and pepper.
Turn up the heat to medium high and add the ½ cup chicken broth. Cook for about 1 minute until the chicken broth has reduced by half. Add the coconut milk and heat through about 2 minutes. Add the chopped tarragon and thyme. Taste for seasoning and add more salt and pepper if needed.
Make a slurry by mixing the remaining 2 tablespoons of chicken stock with the arrowroot or cornstarch. Add to the sauce and cook for about 30 seconds until sauce thickens.
Slice the chicken breasts diagonally in about 1 inch slices and put on a platter. Spoon sauce over the chicken breasts. Garnish with fresh herbs if desired.
Notes
By the way, if you don’t have two large skillets or don’t want to dirty two up, cook the chicken then cover with foil on a plate to keep warm and just use the same skillet for the sauce. Just add a little more olive oil to the pan and continue as above.