There might be a fanfiction where Missy decides to show the 12th Doctor a different side of Gallifrey's history that he was not aware of. This could lead to a whole new adventure for the Doctor as he tries to understand how this new knowledge fits into his view of his home planet.
You can try Archive of Our Own (AO3). It has a large collection of fanfictions, and you can easily search for 12th Doctor and Missy fanfictions there. Just type in the relevant tags in the search bar.
They could also meet in a time - locked situation. The 12th Doctor is trying to solve a mystery from his past, and the 13th Doctor stumbles upon the same situation in her present. At first, they don't understand how they are both involved, but as they start to piece things together, they discover that it's all part of a much larger, more sinister plot. Along the way, they might meet some old friends or foes from their respective tenures as the Doctor. And in the end, they work together to outwit the mastermind behind the plot, all while having some really interesting conversations about their experiences as the Doctor.
In some fanfictions, the 10th and 11th Doctor are brought together by a time anomaly. They are forced to cooperate to fix it, and along the way, they encounter old friends and enemies. The contrast between the 10th's more traditional style and the 11th's more zany approach creates interesting dynamics. They might also have to deal with the consequences of their previous actions in different timelines, which adds an extra layer of complexity to the story.
In the fanfiction, the 11th doctor, with his eccentric and energetic personality, could initially be a bit confused by the 13th doctor's more modern approach. But then they could start to combine their wits. For example, the 11th doctor's knowledge of ancient alien races could be paired with the 13th doctor's understanding of current galactic politics. They could work together to save a planet that is on the brink of destruction due to a complex web of time - related issues. The encounter could also lead to some humorous moments, like when they argue about who has the better sonic screwdriver or who has had the more exciting adventures.
It could be the Daleks. They are a classic Doctor Who villain. They might see the meeting of two Doctors as a threat to their plans for universal domination and try to stop it.
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.
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.