You should save in front of the machine with the Poké Balls in Elm's lab.
When you interact with the machine, you will be able to scroll through all three starters. Here, you can see if they are shiny without having to select them first.
Because you can shiny check all three starters at once without having to select one first, this makes Heartgold/Soulsilver some of the fastest games to do starter reset hunts in. The main downside is that you can't guarantee which starter you will see as shiny first, so it may take a couple attempts to get the shiny starter you're actually looking for.
