  We propose a complete, new formalism to compute unambiguously B-model open
and closed amplitudes in local Calabi-Yau geometries, including the mirrors of
toric manifolds. The formalism is based on the recursive solution of matrix
models recently proposed by Eynard and Orantin. The resulting amplitudes are
non-perturbative in both the closed and the open moduli. The formalism can then
be used to study stringy phase transitions in the open/closed moduli space. At
large radius, this formalism may be seen as a mirror formalism to the
topological vertex, but it is also valid in other phases in the moduli space.
We develop the formalism in general and provide an extensive number of checks,
including a test at the orbifold point of A_p fibrations, where the amplitudes
compute the 't Hooft expansion of Wilson loops in lens spaces. We also use our
formalism to predict the disk amplitude for the orbifold C^3/Z_3.
