I have a player that insisted he could use a basic weapon with one hand and no penality. After I searched his character sheet I couldn`t find any way for him to do that. Is there a trait or quality that allow you to use a basic weapon 1H and not having any malus whatsoever?

Recoil Gloves, or an equivalent system in a suit of powered armor.

Note that this still does not let you use the twin-pistol talents with basic weapons.