Chip a small hole with a pin in the tapered end of the egg. The hole should be as small as possible, but large enough to fit the syringe into.
Insert the syringe into the hole you've made in the egg. Push the plunger down and slowly pull back. This will suck the insides of the egg into the syringe.
Empty the syringe and repeat Step 2. Continue doing this until you have emptied the egg.
Rinse out the egg with water to ensure that all of the yolk and egg white is removed. Allow the egg to dry completely.
Place the towel on a stable surface, and position the egg on the towel.
Pick up your pencil and carefully sketch out the design you want on the egg shell.
Choose an attachment to use with the dremel tool. Typically you will use a wider tip to start with, and then a smaller tip to add detail later.
Turn on the dremel tool and press the tip of the tool onto the egg very gently. Apply just enough pressure to see the mint green layer coming through.
Move carefully along the outline of your drawing and over any other spots you wish to be mint green. Remember to only press as hard as you need to in order to reach the mint green layer. Do not apply more pressure than is necessary.
Decide where you want to add more depth to your design. Very carefully mark these areas with pencil. These are the places where you will be etching to the white layer.
Switch to a finer dremel tip. The white layer is very thin and easily broken through, so you will want to use a more delicate attachment.
Press the dremel very gently on the areas you want to make white. Move slowly and use only as much pressure as is needed to see the white layer.
Continue to add any desired details, such as depth or highlights, in this way.
Sweep away, very carefully, any shell dust from the outside of the egg with the paint brush.