Why do we say "A finger in every pie"?

Meaning:

To be active in many things, often with the implication that the person is overactive to the point of being a busy body

Background:

This old saying presumably originated with kitchen visitors who couldn't resist testing the food by sticking a finger into it, and licking said finger. An early literary reference is...

Lusatia... must needs, forsooth, have her Finger in the Pye

From Parival's Historie of This Iron Age by Bartholomew Harris (1601)

Shakespeare used it in Henry VIII, in which the Duke of Buckingham refers to Cardinal Wolsey, saying...

"The devil speed him! no man's pie is freed
From this ambitious finger".

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