If you activate a brown power that allows you to draw or tuck cards and the deck is empty, you should reshuffle all discarded bird cards to form a new deck.

This is supported by the following rule:

• "If the face-down deck is ever emptied during the game, reshuffle all discarded bird cards to form a new deck."
  (Page 9)

This applies whenever you need to draw or tuck cards and the deck is empty, including when resolving brown powers.

Source: page 9

Source: Core Rulebook (pp. 1, 10, 7, 8, 9)