In a large mixing bowl, combine the 4 cups of shredded green cabbage and 1 cup of shredded carrots and toss them together until mixed evenly.
In a saucepan, combine 1 cup of granulated sugar, 1/2 cup of white vinegar, 1/2 cup of water, 1 teaspoon of salt, 1/2 teaspoon of black pepper, and 1 teaspoon of celery seed and bring this mixture to a boil over medium heat, stirring until the sugar dissolves completely.
Once boiling, remove the saucepan from heat and let it cool slightly.
Whisk in 1 large beaten egg and 1/4 cup of vegetable oil until well combined.
Pour the dressing over the cabbage and carrot mixture and toss thoroughly to ensure every bite is coated.
Allow the coleslaw to sit for at least 30 minutes before serving.