Kacchan Of The Bakugos

The reason Katsuki and Shoto missed each other at each other's doors is likely because of this very moment where they're leaving gifts for one another (and other people).

Katsuki will find something similar to Izuku, actually. A pen, some stationary and some tea as well. Something nice for him to relax with. There's a small letter.

Good morning, Bakugo.
Let's be pen pals. I remember hearing about them back home from an exchange student I met while at a gala. The gala was boring but the idea of getting mail from people was fun. I will write you another letter in the new year to update you on how I'm doing. Please do the same.

Merry Christmas,
From Shoto Todoroki

There is also another notebook in there with a paper slid into it that reads:

P.S. This is for you, too. It may remind you of Izuku, but I think you have good ideas you should put down in it. Things that will help people around you, too. Or you could draw pictures. Or write down names. Denki introduced me to a manga once that revolved around that.

Merry Christmas,
From Shoto Todoroki