Homemade stock is an overnight affair, but the time required will be rewarded whenever you taste a recipe that calls for it.
Place all ingredients except the egg whites (keep these, covered, in the refrigerator) in a large stockpot. Bring to a boil, then simmer 2½ hours, skimming the foam from the top every 30 minutes.
After the 2½ hours, turn off the heat and remove the chicken; reserve for other use. (See Chef’s Tip.) Strain the stock through a fine sieve lined with cheesecloth. Cool to room temperature and then refrigerate, uncovered, overnight.
The next day, when the fat has floated to the top and hardened, skim off and discard. Bring the stock to a boil. While it is cooking, beat the egg whites with a whisk or an electric beater until fluffy. Add to the boiling stock, reduce the heat to a simmer, and stir well. The egg whites will harden and entrap most of the impurities in the stock. (See note below.) Strain the stock again through a fine sieve lined with cheesecloth.
The stock is ready for use now, or it may be frozen for up to a month for later use.
The addition of the egg whites will result in a clearer and more flavorful broth. If you don’t mind a cloudy stock, you may omit this step.
Here’s an idea for the reserved chicken: Remove any bones, and chop the chicken roughly. Mix well with 1 or 2 beaten eggs, and plenty of freshly grated Parmesan cheese and Italian-style bread crumbs. Roll into balls and cook for 10 to 12 minutes in the boiling stock before you add the egg whites. Serve as dumplings in your favorite soup. These can also be frozen for up to a month.