Make the Matzo Balls: In a medium bowl, lightly whisk the eggs. Whisk in the vegetable oil, then the water. In a separate small bowl, stir together the matzo meal, salt and pepper. Stir the matzo mixture into the egg mixture. The consistency will initially be like pancake batter, but it will immediately begin to thicken. Cover and refrigerate for at least 1 hour, or up to 4 hours.
Bring 4 quarts of water to a boil in a large stockpot or Dutch oven over medium-high heat. With moistened hands, form 1 tablespoon of matzo mixture into balls about 1-inch in diameter. Drop the balls into the boiling water so that each falls into the pot in a different place, not crowding each other. When all of the balls are added, reduce the heat to medium-low heat and simmer, covered, for 20 minutes. With a slotted spoon, remove one matzo ball, cut in half and check for doneness. The matzo ball is done when the inside is not dark or wet. If necessary, cook 5 to 10 additional minutes, or until the color is uniform throughout and the texture is light and fluffy (I did end up cooking mine for an additional 10 minutes).
Make the Soup: While the matzo balls are cooking, bring the chicken stock to a boil in a large saucepan. Add the carrots, reduce the heat to medium-low, and cook until tender, about 6 minutes. Use a slotted spoon to transfer the matzo balls to the soup. Stir in the fresh parsley and serve immediately.
Matzo Meal: I use and recommend Manischewitz matzo meal. If using Streit's brand matzo meal, reduce the amount of water to 6 tablespoons.
Larger Matzo Balls: To create large matzo balls about 2½ inches in diameter, use 2 tablespoons of raw mixture and increase the cooking time to 35 to 40 minutes.
Chicken Broth: You can use homemade broth or substitute vegetable broth to make it a vegetarian dish.
Mix-ins/Variations: See the section above about adding proteins, additional vegetables, seasonings, and schmaltz to the soup.
Storing - To store leftover soup, separate the matzo balls from the soup. This will help keep the matzo balls from dissolving into the soup. Place them both in air-tight containers in the refrigerator for up to 5 days.
Freezing - If you are going to save the soup for longer than 5 days I recommend freezing the soup and matzo balls separately. Place the matzo balls on a baking sheet and freeze, then transfer the balls to a freezer ziplock or a freezer-safe container. If you want to skip the pre-freezing step simply place the cooked and cooled matzo balls in a freezer-safe container, without stacking. Freeze the cooled soup in a freezer-safe container.
Thawing - To thaw the soup and matzo balls, move the refrigerator at least 12 hours before you plan to reheat them.
Reheating - Warm the soup in a pot over medium heat with the matzo balls for about 7-10 minutes.