Bring water to boil in a medium saucepan. Add broccoli; cook for 2 minutes. Add noodles (reserve the seasoning packet); cook until the noodles are tender, about 3 minutes more. Drain the broccoli and noodles and return to the pan.
Add sesame oil, sesame seeds and half of the seasoning packet (discard the remainder or reserve for another use); toss well to combine. Transfer to a bowl; top with egg and serve.