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I am definitely not recovered 'a few mins' after finishing a TT - which is the absolute test of what your FTP is. My HR might be back to a manageable level quick enough, but the rest of me wants to roll into a ditch and cry for a bit. Half an hour later it was all a fantastic idea, I had a great time and will definitely go faster next time.
Race efforts are always going to push you a little further than training, just because you're motivated. In TT's find that no matter what my plan is, I always go out harder than I meant to and just hang on (whilst hating myself, the world and everything to do with cycling). Maybe one day I will learn, but I'll probably go slower. It's really really difficult (impossible) to replicate those efforts in training when the clock isn't ticking and you don't have someone faster than you 1min behind you.
Not sure what you mean by 'keep going'. You're recovered in a few mins, then spin home?