Heat the oil over medium heat in a frying pan or casserole with a lid.
Grate in the garlic.
Cook, stirring constantly, for a moment or two.
Stir in the spring onions and then put the chicken breasts in the pan.
Cook for 5 minutes, then turn the chicken breasts.
Cook for 5 minutes more.
Add the vermouth or white wine, dried tarragon, a pinch of salt and a twist of freshly ground black pepper.
Turn down the heat, pop the lid on the pan and cook for 10 minutes. Check half way through cooking to be sure the chicken isn't sticking. Gently dislodge it if it is, and adjust the heat.
Check that the chicken is cooked through. It should reach an internal temperature of 165 degrees Farenheit (75 degrees Celsius). I use a meat thermometer to be sure.
When the chicken is cooked, remove it to a warmed clean plate and cover with foil.
Turn the heat back up on the remaining liquid in the pan, and bring it to the boil.
Add the cream and most of the fresh tarragon, stirring well.
Taste carefully (it's hot!) and add salt and pepper to taste.
Serve the chicken breasts on warmed plates with some of the sauce poured over. Garnish with the remaining fresh tarragon.