Preheat oven to 325, then cook the onions and carrots on the stovetop (in a casserole just large enough to hold the whole chicken) in 2 T butter until vegetables are tender but not brown.
Rub the outside of the bird with 1.5T of the butter and place the other 1.5 T inside the bird. Sprinkle the outside with half of the salt and tarragon and the inside with the other half. Truss and arrange the chicken breast-side-up on top of the sauteed vegetables in the casserole.
Pour the wine into the casserole and add enough of the chicken broth to reach about 1/3 of the way up the chicken. Add the herbs and giblets. Bring to a simmer on top of the stove. Drape the damp cheesecloth over the breast and thighs. It should be long enough to fall into the liquid on either side so that is will draw the broth over the chicken and basted it during the cooking. Smear remaining butter over cheesecloth and top with a layer of waxed paper. Cover casserole and set in the preheated oven.
Bake chicken for 1 hour and 40 minutes. Watch to make sure that the liquid remains at a very soft simmer and adjust temperature accordingly if needed. When chicken is done (juices should run clear), remove chicken to a platter and strain the cooking juices. Return one cup of the stock to the casserole with the chicken. Cover to keep warm while you prepare the sauce.
For sauce: Melt butter in saucepan, blend in flour with a wooden spoon and stir over moderate heat until butter and flour foam together for 2 minutes without browning. Remove from heat and as soon as it stops bubbling, add 2 cups of the hot chicken stock,, blending vigorously with a wire whisk.when smooth, set over moderate heat and stir slowly as sauce thickens and comes to a boil. Boil, stirring for 2 minutes. Sauce will be quite thick.
Pour in 1/3 cup of the cream and add successive spoonfuls until sauce thins out but is still thick enough to lightly coat a spoon. Check seasoning and add salt, white pepper and lemon juice to taste. Just before serving beat in the remaining 2 T butter.
Carve the chicken and top with the sauce.