Perhaps they find themselves in a situation where they have to rescue a group of people trapped in a parallel universe. Jack's skills as a former Time Agent and the Doctor's ability to navigate difficult situations would be crucial in this rescue mission. It could also be a chance for them to learn more about each other's pasts and grow as characters.
It's possible that their encounter would be a more serious and emotional one. Jack has seen a great deal of loss and the 13th Doctor also has her own emotional baggage. They could bond over shared experiences, like the loss of friends or the struggle to protect the universe. Jack might offer some unique perspectives based on his long life and the Doctor could in turn show him new ways of looking at things.
One possible plotline is that they meet in a time - locked situation. They have to work together to break the time lock. Another could be that they meet because of a shared enemy that has somehow found a way to target both of their timelines. They would then have to put aside their differences and use their individual strengths to defeat the enemy.
Another plotline could be that the immortal has some knowledge that the Doctor needs. The Doctor is constantly seeking knowledge, and the immortal, having lived for so long, might hold the key to solving a big problem in the universe. So the Doctor has to convince the immortal to share that knowledge, which might not be easy as the immortal could be very guarded about it.
One possible plotline could be that the Doctor and Jack Harkness team up to save a planet from an unknown alien threat. They use their combined knowledge and skills to outwit the aliens.
They could meet during a time-traveling adventure gone wrong, getting stranded in the same era.
A storyline could involve a character from Torchwood getting trapped in a time loop. The 11th Doctor shows up and, with his knowledge of time mechanics, tries to break the loop. Along the way, he gets to know the Torchwood team better and they learn a great deal from him about the nature of time and the wider universe.
One possible plot twist could be that they find out they are both parts of a larger plan by an unknown enemy. Maybe an ancient force has been manipulating their timelines to bring them together for some sinister purpose.
In fanfiction, when the 13th Doctor meets the 11th Doctor, there could be a lot of chaos and fun. They might exchange stories about their adventures. Maybe they would team up to fight a common enemy that spans across their different timelines.
In fanfiction, when the 11th Doctor meets the 12th Doctor, there could be a clash of personalities at first. The 11th is so full of child - like wonder and the 12th is a bit more reserved. But then they might find common ground in their love for the TARDIS and for saving the universe. They could end up learning from each other. For example, the 11th Doctor might teach the 12th how to be more carefree sometimes, while the 12th Doctor could show the 11th how to be more strategic in dangerous situations.
You can start by looking on Archive of Our Own (AO3). It has a large collection of fanfictions, and you can easily search for 'Jack Harkness x Doctor' there. Another place could be Fanfiction.net, although the interface might not be as user - friendly as AO3. There are also some Doctor Who - specific fanfiction forums where you might find some great works about Jack and the Doctor.