In order to defeat Brodie, you need to inflict any amount of damage with two separate attacks.
The first wound will remove one of the cat's initiative cards, the second wound will remove the other initiative card.
Note to clarify: If your attack would deal more than one wound, you cannot place a wound marker on both initiative cards - you target only one initiative card and any excess wounds are ignored.
See page 11 in the rulebook:
Wounds on boss minions
Boss minions normally have multiple initiative cards on the initiative track. Each time a mouse inflicts a wound on that boss, choose which initiative card to place the wound on. When a boss initiative card has
a number of wounds on it equal to or greater than its life value, remove that initiative card from the initiative track. Once the last initiative card for a boss has been removed from the track, remove its figure from the board as well - it has been defeated.