With nick mastery any character gets a free dagger throw with an attack.
Bit of a click bait on the title, but please hear me out and check the math.
We get a free object interaction per turn.
Nick allows you to get the extra attack of a light weapon as part of your normal attack, but gives little specification to order of operations and which weapons need to be light.
Therefore this seems plausible.
Stow current weapon. Pull out the dagger and throw it (free draw as well as free attack). Attack an enemy with the weapon you previously stowed (free draw). Continue doing whatever you wanted to do.
This seems kind of silly but at the cost of a weapon mastery “slot” and it being necessary you attack, is anything stopping you from doing this?
Bonus question because this has been somewhat answered (at least one other light weapon attack needs to be made for this):
I have a spear (not light) in one hand, my other hand is free. I am a level 5 barbarian with extra attack. Can I throw two daggers then attack another enemy with my spear? This seems to work, even if it's kind of dumb.