1. **Prepare the Veggies:** Finely chop the carrots, green onions, and any other mix-ins. It is crucial to chop them very small so the omelette rolls smoothly without breaking.

2. **Whisk:** Crack the eggs into a large bowl (preferably one with a pouring spout). Add the milk, salt, and pepper. Whisk thoroughly until the yolks and whites are completely combined and no streaks remain. Stir in the chopped vegetables and ham.

3. **Heat Pan:** Heat a non-stick skillet (a square tamagoyaki pan is traditional, but a round 10-inch skillet works too) over medium-low heat. Brush the pan lightly with oil or wipe it with an oiled paper towel.

4. **First Layer:** Pour about ¼ of the egg mixture into the pan to cover the bottom thinly. Swirl the pan to coat evenly.

5. **Roll:** When the egg is set on the bottom but still slightly wet on top (about 2 minutes), gently lift one edge with a spatula and fold it over about 1 inch. Continue rolling the egg up until you reach the end. Push the rolled egg to the side of the pan.

6. **Second Layer:** Brush a little more oil on the empty part of the pan. Pour another ¼ of the egg mixture into the empty space. **Important:** Lift the cooked egg roll slightly so the raw egg flows underneath it—this bonds the layers together.

7. **Repeat:** Once the new layer is mostly set, roll the cooked log over the new egg. Repeat this process until all the egg mixture is used.

8. **Shape & Serve:** Remove the finished log from the pan and let it cool on a cutting board for 5 minutes (this helps it hold its shape). Slice into thick, even rounds and serve.