Heat the oil in a large skillet, saucepan or casserole over a high heat. Season the chicken breasts generously with salt and pepper. Place the chicken breasts in the pan and cook for 4 to 5 minutes, or until they are golden and caramelised, before turning them over and cooking them for another 4 to 5 minutes. The best way to achieve a nice colour on the chicken is to not move the chicken too often.
Once the chicken breasts are golden and caramelised on both sides, remove them to a plate. The chicken does not need to be fully cooked through at this stage.
Add the shallots (or onions) and garlic to the pan and cook on a medium heat for 3 to 4 minutes, stirring frequently, until they have softened.
Pour the white wine into the pan and let it simmer for 2 to 3 minutes until it has reduced slightly. Use a wooden spoon to stir the mixture and to lift any caramelised bits from the pan.
Remove the pan from the heat and pour in the chicken stock and cream, together with the fresh tarragon. Stir to combine, then return to the heat and let the sauce simmer gently for 4 to 5 minutes, stirring occasionally.
Taste to check the seasoning and add more salt and pepper, as needed.
Increase the heat to a fast simmer, then slowly pour in the cornflour slurry, stirring the sauce as you do. You may not need all of the cornflour and water mixture – use as much as you need until the sauce has thickened slightly.
Return the chicken to the pan, turn the heat down to low, and cover with a lid. Cook gently for 10 to 15 minutes, or until the chicken breasts have cooked through (I use a meat thermometer to check they’re cooked to 74C).
Garnish with parsley or more tarragon before serving. It’s delicious with roast potatoes or mashed potatoes, plus your favourite veg. It’s also delicious with pasta (see Notes).