A manga like otonari no tenshi sama.
But mc is the one who save the fmc.

They live as neighbour on apartment. Someday mc see the girl have fever and fall in front of the door. Because he didnot want to search the key from the girl uniform, mc take the girl to his home.
Mc can cook because his parent have a french restaurant.

Fmc want to pay mc for taking care of her when she sick.

Idk if you're still looking for it, but it's probably "How to Melt the Ice Lady". It's a novel though, not a manga. Sorry if i'm wrong, it's been a while since i last read it.

There's probably manga with that kind of plot and i just happened to not know about it. The only title that crossed my mind with that kind of plot was the novel.