Generally, you should design the whole puppet first so the scale and look of both the head and the beak are correct. Typically, for a bird, I would pattern the skull and the two halves of the beak as three separate pieces. The two halves of the beak are attached via the mouth plate and the top of the beak should be attached to the skull. You probably want the bottom of the beak "free floating" (attached to the rest of the puppet's structure only via the mouth plate) so that it can open and close freely, but it really depends on the design.