  We give a new construction of the Brownian annulus based on removing a hull
centered at the distinguished point in the free Brownian disk. We use this
construction to prove that the Brownian annulus is the scaling limit of
Boltzmann triangulations with two boundaries. We also prove that the space
obtained by removing hulls centered at the two distinguished points of the
Brownian sphere is a Brownian annulus. Our proofs rely on a detailed analysis
of the peeling by layers algorithm for Boltzmann triangulations with a
boundary.
