In a large bowl, whisk together the eggs and 1/3 cup cold water. Add the rendered or reserved chicken fat and whisk until the fat blends in. Mix in the salt and pepper. Gradually but quickly stir in the matzo meal with a spoon; the mixture will be thick and stiff, like muffin batter. Don't overmix. Chill for at least 1 hour or up to 3 hours.
Line a baking sheet with parchment or waxed paper and fill a bowl with cold water. Dip a large soupspoon in the water, and gently scoop up the chilled matzo mixture and shape it with your hands into 12 medium balls (about 1-3/4 inches in diameter) or 18 smaller ones (about 1-1/4 inches in diameter), being careful not to compact them. Put the matzo balls on the lined baking sheet. Cook immediately or refrigerate for up to 1 hour.
To cook the matzo balls, bring 1 or 2 large pots of salted water to a boil. Drop in the matzo balls, cover the pots, and reduce the heat after the water returns to a boil. Simmer, covered, until the matzo balls have doubled in size and have lightened all the way through (cut one in half to check) 30 to 40 minutes; drain. Cooked matzo balls can be held at room temperature for several hours.
To serve, bring the chicken broth to a boil. Taste for salt and pepper. Add the matzo balls and heat until they're hot in the middle, 8 to 10 minutes. With a slotted spoon, put 2 medium or 3 small matzo balls in a warm soup bowl. Ladle in hot broth and sprinkle generously with the parsley. Serve right away.