Kids (2008)

Even though the power is different (Asato can transfer injuries to himself and others), it's a heartwarming film focusing on the relationship between the friends and each characters personal development.

In both of these, the main leads share a similar relationship based on mutual help and understanding. If you loved the close bond between the two main characters in Aozora no Tamago, then you'll also love the heartwarming atmosphere from Kids, even though one is a drama and the other one is a movie.

Both dramas Koike Teppei acts and has a supernatural ability.
+There's also bromance in both of them <3

Both explore the consequences and morality of changing the natural order of pain, life, and death.

In both films, a young man has a supernatural power that allows him to transfer wounds from other people onto himself.