Combine all ingredients in a large pot over high heat; bring to a boil. Reduce heat to low and simmer, uncovered, for 4 to 5 hours. Midway through, add water to replenish to original liquid level.

Strain out solids with a colander set over another large pot or bowl. (Pick through the solids for the chicken meat and skin, carrots, celery and garlic, discarding the bones, herbs and onion. This intensely chicken-y mélange can be rationed out to a dog - about 3 cups food, more if you add some cooked rice.)

Pour stock through a fine mesh sieve back into the first pot. Cool to tepid. Refrigerate overnight in the pot or transfer to freezer-safe containers.