Break down the leftover rotisserie chicken carcass into smaller pieces. Add any leftover skin or chicken pieces for extra richness.
In a large stockpot, add the chicken carcass, onion, carrots, celery, garlic, sage, thyme, bay leaves, and black peppercorns. Pour in apple cider vinegar and kosher salt.
Add enough water to fully submerge the ingredients, about 8 cups.
Place the pot over medium heat and bring to a gentle simmer. Skim off any foam or impurities that rise to the top within the first 30 minutes. Reduce to low heat and let simmer uncovered for 4-6 hours.
Remove the pot from heat and strain the stock through a fine-mesh strainer or cheesecloth into a large bowl. Discard the solids.
Let the stock cool to room temperature, then refrigerate for several hours or overnight. The fat will solidify on the surface—skim it off if desired. The stock should be gelatinous when chilled.