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Right after ‘The Doctor Falls’ aired, and a part of the fandom - including me - nearly lost their minds because the Doctor’s flashback included Clara, I saw someone - I really can’t remember who, I’m so sorry - comment in their tags that the choice of episode seemed odd. The snippet of Clara yelling ‘Doctor?’ was from ‘Last Christmas’ - why was that?
‘Last Christmas’ takes place in a dream, right? Well, most of it, except the very last scene where the Doctor and Clara basically run away together on Christmas night. Thing is, we don’t always remember dreams fully. We suddenly remember scenes, colours, wisps of emotion that we’d felt in the dream, like deja vu. Other times they lie submerged in our minds, waiting, breathing. The Doctor did say that chances of any of them remembering the dream after they’d woken up were ‘very slim’, but not non-existent. He and Clara did remember their dream after waking up.
The episode itself was all about mourning and hope entwined, about impossible second chances and finding each other again across time zones, across many worlds and fantasies. It was about discovering a space for more stories where there didn’t seem to be any hope for it. Also, in that particular scene, the Doctor is stomping away, simply not caring that the dream crabs are still there on Earth, still a grave danger to people. Clara pulls him back, unimpressed by his carelessness. She asks him the big question, triggering his own brainwave. She was doing what he trusted her to do best, what she did even as she went on her way to her own death, after coming back and leaving him again. She was calling on him to be the Doctor.
So then this is how the Doctor remembers Clara, after forgetting her. Deep inside his own mind, he still mourns her. There is still that hurt from losing her, but it’s not an uncontrollable pain. He’s following Clara’s last wish for him while still hoping, without possibly even knowing it, to see her again. It also reflects the nature of his memory of Clara - a phantom, like that small sad drift of her song in ‘The Pilot’, in a polar base that never existed, preserved in ice and snow, rooted in desire and faith rather than hard facts. Yes, it’s just a dream, but why should it mean it’s not real?