Recap / Doctor Who 2005 RBS "Attack of the Graske"
"I need your help! Do you fancy a trip in the TARDIS?"

"You've been watching my adventures, and I've been watching some of yours. Including you, mate, where do you get the energy?"
The Doctor actually has a word with his audience for the first time in decades!

The one where you are the companion!

"Attack of the Graske" is a 2005 BBC Red Button Interactive special, broadcast right after "The Christmas Invasion". Available on the BBC website.

It's a fourth wall breaker special, much like "A Fix with Sontarans" before it, so draw your own conclusions as to its canonicity.
The Doctor has dropped off Rose at an ABBA concert, and since the Earth is being attacked by a Graske, he needs a temporary companion. An Adventurer Is You!

The Graske first terrorizes a regular English family during their Christmas celebration, then hides out in Victorian London for a bit. The Doctor connects his sonic screwdriver to your remote and asks you to help him track the creature down. But first, you get to help him fly the TARDIS.

You eventually sneak into the Graske base, unlock its doors, and find a lot of Human Popsicles (and some alien popsicles) in pods. One of them, a Slitheen, breaks free and starts chasing the Graske, and you have to make a choice... do you teleport all the people back home, or do you freeze them all?

Once you've saved the day, the Doctor happily thanks you, and tells you he might even pick you up as a proper companion someday. (Also, he warns you not to switch to ITV. The universe might implode.)