Combine 3 cups water, tea leaves, soy sauce, black soy sauce, zest, cinnamon stick, star anise, and ¼ teaspoon salt in a large saucepan; bring to a boil. Reduce heat, cover, and simmer for 3 hours.
Meanwhile, place eggs and remaining 1 teaspoon salt into a medium pot; cover with cold water. Bring to a boil, reduce heat, and simmer for 20 minutes. Remove from heat, drain, and cool. Tap cooled eggs with the back of a spoon to crack shells; do not remove shells.
Remove the saucepan from the heat. Add eggs and let steep for at least 8 hours, or for a richer flavor, up to 1 ½ days.
