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.
Does the mount recover HP from this attack when it delivers it, or does the caster?
Relevant Rules Text;
PHB, Page 202; Rules of Magic -> Range: "Other spells, such as the shield spell, affect only you. These spells have a range of self."
PHB, Page 204; Rules of Magic -> Targets: "A typical spell requires you to pick one or more targets to be affected by the spell’s magic."
Vampiric Touch; Range -> Self
Valid Targets: Self. "Affect only you."
Conclusion; the attack roll is a feature granted by the spell cast on yourself, with its own separate targeting function, much like the natural weapons function of Alter Self.
Alternate conclusion; Vampiric Touch (and Alter Self; natural weapons) does nothing, because creatures other than self are not a valid target for any component of the spell.
The answer to the suggested and linked question was asked separately explicitly because of the spell text saying "you regain," as opposed to "Make a melee spell attack," without being preceded by "You." There is room, I feel, for the answers to not cleave the same way.