In public play, I have seen multiple wizards cast Vampiric Touch to deliver the spell through their familiar. I am curious than, how it works in terms of the action economy.
Vampiric Touch, PHB pg 285
Make a melee spell attack against a creature within your reach. On a hit, the target takes 3d6 necrotic damage, and you regain hit points equal to half the amount of necrotic damage dealt. Until the spell ends, you can make the attack again on each of your turns as an action.
Find Familiar, PHB pg
Finally, when you cast a spell with a range of touch, your familiar can deliver the spell as if it had cast the spell. Your familiar must be within 100 feet of you, and it must use its reaction to deliver the spell when you cast it. If the spell requires an attack roll, you use your attack modifier for the roll.
Does this mean I can cast Vampiric Touch once, and the familiar can deliver it round after round as a reaction, while I continue concentrating, avoiding the need for me to use an action? If not, how does it work, and do I need to "recharge" the familiar?