Hey Anon, I’m not sure on this again. I don’t really remember any stories where Tom kills the basilisk for Hermione but the second half of your ask sounds like maybe Somewhere in Time? -JD
Somewhere in Time by SerpentInRed
M | Complete | 345k
Sent back in time by a mysterious person and trapped in the past with a missing Dumbledore and an overbearing, charismatic Dark Lord, they had no idea how much they could dabble with time before the world they had known shattered into pieces.
He says, easing his way to flip onto his hands. He keeps himself upright for at least 5 seconds before getting back onto his feet.