Is electrical induction achieved more efficiently by coiling a wire than using a cylinder?
Faraday's law of induction says simply: -

Induced voltage = \$N\dfrac{d\Phi}{dt}\$ where N is the number of turns.

So, if you have many turns you get more voltage. This is usually the main motivator for winding many turns but, one turn is fine and can be just as good (and sometimes better) if the frequency is high enough because notice that \$\dfrac{d\Phi}{dt}\$ tells you that induced voltage is also proportional to frequency.

However, one further important fact is that a solid sheet conductor in a varying magnetic field will be very inefficient due to eddy currents induced in the sheet. This is why transformers use insulated laminations or ferrite (low conductive materials) in their construction.

These eddy currents sap the field and reduce induction.