The Man Who Lived
For @superrpowerlesshuman. You didn’t ask for it, but you mentioned it so it’s yours now. ;)
Hear me out. Imagine that the Tardis is a die-hard romantic. So part of her mission is to help soulmates cross oceans of time (shout-out to Dracula) to find each other. The Doctor’s Companions aren’t just there because he likes them, but because the Tardis picks them once she has a match.
So Karen, backpacking through Europe to avoid dealing with family issues, stumbles across the Doctor and off she goes on the adventure of a lifetime. However, it becomes obvious they keep running into different incarnations of the same man, over and over again.
It doesn’t register the first time. The second time...well, it’s a coincidence explained as a quirk of time. The third time? That’s a pattern, and it intrigues even the Doctor.
Frank Castle is a mystery worthy of the Doctor’s attention. Also, the Tardis is getting pissed off because these soulmates are stubborn and fighting against the inevitable. So she has to keep throwing them together repeatedly and they just need to kiss already since she’s getting bluer each trip.
Just sayin’ it’s possible.