What fillings to put into an omelet is a decision that can really bring out an aspiring chef's creativity. So while this classic combination is great for first-timers, once your child gets the hang of omelet making, encourage her to experiment with other ingredients.
Wash the broccoli florets and cut them into bite-size pieces. Place them in a small saucepan and add just enough water to cover the bottom of the pan. Cover the pan tightly and heat the broccoli over high heat, steaming it for 2 to 3 minutes, until crisp-tender. Drain the florets in a colander, being careful to avoid the steam, then set them aside to cool.
Crack the eggs into a small bowl. The trick is to achieve as clean a break as possible by striking the egg against the bowl's rim with just the right amount of force. Too little, and you'll end up with small shell fragments; too much, and you'll smash the entire shell. If you happen to get any shell fragments in the bowl, use a clean teaspoon to remove them.
Add the water and salt to the eggs and use a fork to beat the mixture just until blended. Now, before you do anything else, get the rest of your breakfast ready (pour your orange juice, put your bread in the toaster, and so forth), because an omelet cooks in 2 minutes or less from start to finish.
Put your pan on a front burner with the heat set at slightly higher than medium. If you're right-handed, point the handle toward your left side; if you're left-handed, point it toward your right. Let the pan heat for 2 to 3 minutes so that it gets fairly hot. Put a tablespoon-size pat of butter in the middle of the pan and let it melt. If the butter immediately starts to sizzle and smoke, your pan is too hot. Remove it from the heat to cool for a minute or so before proceeding. Tilt the pan to spread the melting butter over the entire bottom.
Pour the beaten eggs into the prepared pan and wait a few seconds for the bottom of the eggs to cook just a little. Then use a wooden spoon or spatula to start gently pushing the eggs away from the edge of the pan toward the center. As you do so, tilt the pan so that uncooked egg runs into the empty spot you've created. Work your way around the edge in this fashion until the egg is no longer runny.
Using the back of a wooden spoon or spatula, spread any remaining uncooked egg across the surface so that it will continue to cook evenly.
When the surface of your omelet looks just a little undercooked, it's time to add your fillings (the egg will finish cooking as you complete the next 2 steps). Arrange the broccoli over the half of the omelet farthest from you. Sprinkle the cheese and the ham over the broccoli.
Holding the pan handle with one hand, slide a spatula under the plain half of the omelet and fold it over the filling in one steady motion.
Using an underhand grip, tilt up the pan handle and slide the omelet out of the pan and onto your plate. Bon appétit!