This recipe makes a nice savory tamago maki. I've put it together using various bits of advice and a lot of experimentation. Moreover, I've made it without a tamagoyaki pan. It can also be put on top of nigiri (rice balls) to make tamagozushi.
Beat the eggs, strain through fine sieve, and add to dashi stock, mirin, sugar, and soy sauce.
Heat pan (tamagoyaki pan OR a small-ish skillet--a flat, square pan would work best, but I have used a round one with little difficulty) to medium high heat with PLENTY of cooking oil. I find that the oils sold in spray bottles, like Pam, work the best to spread it out evenly.
Pour enough of the egg mixture onto the pan to spread evenly over the surface, about 1/3 of the mixture.
Allow mixture to cook just until the egg is set (but not cooked all the way, or the layers won't stick to each other when you roll it). I determine this by looking for the dark yellow, orangish color to appear. Sprinkle some of the green onions over this.
Start rolling the omlette by picking up the edge farthest away from you and pulling it toward you. I often do this in steps, flipping about 1/4 of the edge over, letting it set, then flipping that over (like a jelly roll) etc, until the entire omlette is rolled up.
Now, re-oil the exposed part of the pan and add 1/2 of the remaining mixture. Repeat steps 4 and 5.
Repeat step 5, which should use up the rest of the egg mixture.
Allow to cool (so that it cooks all the way through) and wrap in paper towel. You can make the shape more rectangular (for tamagozushi, for example) by gently putting pressure on the paper towel.
Cut into slices (usually 0.25-0.5 cm or so) to serve. Garnish with remaining green onions.