How can I be sure my food is ready for a raw egg to be dropped on top?
There is some slight confusion but there are two methods for this. You either cook the egg separately, which is how they make BiBimBop or you poach or bake the eggs if you cook the egg on top of your food. With the latter method, the eggs generally cook very quickly so it doesn't take a lot of time but you are still cooking the egg using a heat source, you are not relying on the food to cook the egg for you. 

When you make BiBimBop using a whole egg, you fry the egg in a pan before you place it on top of the food. This is why the egg often has a bit of browning around the edges of the white.

Here's an image of a BiBimBop with a whole egg:



There is a "how to make BiBimBop" video here that shows (around 8:50 in) that you cook the egg separately and place it on top of the finished bowl.

The other option is to bake or poach the egg by placing them on top of the food (which is still on the stove) and then covering the pan (or putting it in the oven) to essentially steam the egg, which is often used for hash. This gives a different final product than frying the egg and placing it on top, as the steam will cook the top of the egg while the still-cooking food heats the bottom.

Here's a video of Venison Hash where they use this method at around 5:05.

And here's a recipe that uses the oven version of this method.