[Borek]

I₂ + 6HNO₃ -> 2HIO₃ + 6NO₂ + 2H₂

I am not asking about the result, I am asking HOW. I can balance it by inspection, I can balance it using oxidation numbers, but I can't balance it using half reactions. Do I have a senior moment, or is it some deeper problem?

[AWK]

In the presence of concentrated HNO₃ an elemental H₂ cannot exist.
Half reaction method needs two elements changing oxidation number on both sides.
But even that your reaction can be balanced by half reaction method using oxidation number 0 for hydrogen on both sides

[Borek]

I can't see it 

[AWK]

Change H from HNO3 to neutral D and treat D as spectator atoms.
Than you have
I₂ + NO₃ = IO₃ + NO₂ (all neutral molecules)