Enjoy the soup as is, or garnish with parsley leaves.
To prepare chicken soup, place chicken in an 8-quart stockpot; add 3 quarts water. Bring to a simmer, skimming foam from surface. Adjust heat to maintain simmer (boiling will turn soup cloudy).
When no more foam appears, add carrots and next 6 ingredients (through squash). Simmer, uncovered, 3 hours; add herbs during the last 15 minutes.
Cool slightly, and strain through a fine sieve. Discard solids. Cool soup to room temperature; cover and refrigerate overnight. Scoop off congealed fat, reserving 3 tablespoons.
Place eggs in a medium bowl; whisk to blend. Stir in matzo meal, 1/4 teaspoon salt, 3 tablespoons reserved fat, and 1/4 cup soup. Cover; chill 2 hours or until firm.
Bring a wide pot of water to a boil. Gently but thoroughly roll 1 rounded tablespoon dough between moistened palms into a ball; place on a plate. Repeat to form 16 balls total.
Maintain water at a gentle boil. Gently add matzo balls; cover and simmer 35 minutes or until tender and slightly al dente, stirring after 10 minutes.
Bring 6 cups soup and remaining 3/4 teaspoon salt to a simmer (reserve any remaining soup for another use). Add matzo balls to soup using a slotted spoon. Simmer, uncovered, for 10 minutes.
Adapted with permission from The Seasonal Jewish Kitchen, copyright 2015 by Amelia Saltsman, Sterling Epicure, an imprint of Sterling Publishing Co., Inc.