Place chicken carcass in large stock pot. Add carrots, celery, onions, and garlic around the chicken.
Sprinkle oregano, parsley, thyme, peppercorns, salt, and bay leaves on top.
Add 12 cups water. Bring just to a boil, then immediately reduce to a gentle simmer.
Simmer 3½-4 hours, stirring occasionally.
When meat is falling off bones, remove from heat.
Carefully strain contents through fine mesh strainer to separate liquids and solids.
Reserve the liquid (this is your stock!). Pick any larger chicken pieces from strainer and save separately for soup. Discard remaining solids.
Refrigerate up to 4 days or freeze up to 3 months.