Heat the garlic oil in a frying pan or Dutch oven that has a lid and in which the chicken breasts will fit pretty snugly. Add the scallions, stir, then sprinkle in the freeze-dried tarragon, stir again and cook them in the garlic oil for a minute, stirring some more as they cook.
Put the chicken fillets into the pan, curved side down, and cook for 5 minutes, watching the scallions don't burn. If they look like they're beginning to, scrape them from the pan and let them sit on the chicken pieces.
Turn over the breasts, and add the vermouth (or white wine). Let the vermouth bubble up, then add the salt. Put the lid on, turn the heat down low and leave it to simmer gently for 10 minutes. Check the chicken is cooked through by making a small cut into the thickest part and ensuring the juices run clear - if not, simmer for a few minutes longer and check again.
Remove the chicken breasts to warmed plates. Bring the remaining liquid to a boil, add the cream and stir well, then sprinkle in the fresh tarragon, stir again and give a good grind of white pepper.
Pour the sauce over the chicken breasts, and give a final scattering of tarragon to serve.