Cook pasta or rice to serve with chicken, set aside.
Heat a 2-3-inch deep frying pan or dutch oven over medium high heat* and add olive oil to pan. Generously salt and pepper chicken on both sides.
Carefully place chicken in pan with olive oil. Cook for one minute on each side. The chicken should start to turn golden in color. Remove to a plate or platter. Set aside.
Turn heat to medium. Do not clean out pan. Add 2 tablespoons butter to the pan.
Add 2 tablespoons flour. Cook and whisk just until flour and butter are incorporated, scraping up browned bits from bottom of pan. The mixture will most likely be a little lumpy. Add fresh lemon juice to the pan.
Slowly add 1 ½ cups of chicken broth, whisking while adding broth. Add 1 cup of half and half. Add tarragon, salt and pepper to taste.
Heat until smooth and bubbly. Add chicken back to the pan with the sauce and cook for 5 minutes over medium heat. The sauce should gently boil while cooking the chicken. If it is not a gentle boil, adjust the heat. Cover pan with lid and remove from heat after cooking and let sit, covered, for 5 minutes.
Just before serving, garnish with lemon slices and freshly chopped parsley. Serve with rice or pasta while hot.
