Brown the Chicken: Place the whole stewing hen, breast side down, into a large stockpot with 1 to 2 tablespoons of ghee. You can sprinkle a bit of salt and pepper into the stockpot along with the ghee (and a bit of red pepper flakes for spice, if desired). Over medium heat, let it gently brown until the skin turns a light golden color. Carefully flip the chicken and brown the back side as well.
Deglaze with Vermouth: Pour in 1 cup of very dry (white) vermouth and stir to loosen any browned bits from the bottom of the pot. Let it simmer for about a minute to allow the alcohol to cook off. If you prefer not to use vermouth, simply deglaze with water and apple cider vinegar.
Add Water and Simmer: Add enough cold water to just cover the chicken. It's okay if part of the chicken floats above the water. Bring everything to a boil, then reduce to a gentle simmer, skimming off any foam or impurities that rise to the surface.
Add Aromatics: Once you have skimmed the impurities off the top, add the carrots, celery, onion, bay leaf, and peppercorns.
Slow Simmer: Reduce the heat to the lowest setting, cover the pot, and let it simmer on low for 3–4 hours until the chicken is falling off the bone.
Strain and Store: Remove the chicken and vegetables, then strain the broth through a fine mesh strainer lined with a flour sack towel or cheesecloth into a clean bowl.
If you want to defat the broth, use a fat separator. However, be sure to keep the chicken fat (schmaltz). Don't discard it! You can use it to make chicken livers or to sauté vegetables or cubed potatoes.
Serve the chicken along with the broth, or reserve the chicken for a separate recipe.