Grammar: Gerund

1- The best thing to do is to write a note, and asking him to authorize me.

No 'and'. This is one choice, a very natural one, Fargh. Another could be,

The best thing to do is to write a note, in which you will ask him to authorize me.



2- The best thing to do is to write a note (and) to ask him to authorize me.

Again, no 'and'.

Maybe grammatical but they don't quite make sense since it isn't clear how or whether the writing of the note is connected in some way with the authorization

Somewhat OT Fargh but my take:

The best thing to do is to write a note asking him to authorize me

Since the "do's" get so alliterative I might instead have writ:

The best thing is to write a note asking him to authorize me, or

Best we write him a note requesting my authorization