In a large shallow roasting pan place chicken pieces, carrots, and onion. Bake, uncovered, in a 450 degree F oven for 30 minutes or until well browned, turning occasionally. Drain off fat. In a 10-quart Dutch oven or kettle place the browned poultry, carrots, and onion. Pour 1/2 cup water into roasting pan, stirring to scrape up browned bits. Pour this liquid into Dutch oven.
Add cabbage, celery, tomato, peppercorns, parsley, bay leaf, salt, thyme, garlic, and water. Bring to boiling; reduce heat. Cover; simmer for 5 hours.
Strain stock by ladling it through a sieve lined with 1 or 2 layers of 100% cotton cheesecloth; discard meat, vegetables, and seasonings.
Clarify stock, if desired. To clarify, separate egg, crushing and reserving the shell. Stir together 1/4 cup cold water, the egg white, and the crushed eggshell. Add to stock; bring to boiling. Remove from heat; let stock stand for 5 minutes. Strain clarified stock again through a sieve lined with 1 or 2 layers of cheesecloth.
If using stock while hot, skim fat using a metal spoon. Or, ladle stock into pint or quart jars or freezer containers; cover and chill quickly; when cold, lift off fat with a fork. Stock may be stored in the refrigerator for 3 days, or in the freezer for up to 6 months. Makes about 8 cups (8 servings).