  In this work we discuss the application of the Hamilton-Jacobi formalism on
the scalar field implementation of Generalized Chaplygin Gas models. This
corresponds to a Generalised Born-Infeld action for the scalar field, which in
an initial fast-rolling phase mimics a matter-like behavior and in the final
slow-rolling phase mimics a cosmological constant. In order to enrich the
phenomenology of the model, we add an extra functional freedom, specified
through a scalar potential for the field. Interestingly, we find that, due to
the lifting induced by the non-standard kinetic term, an asymptotic de
Sitter-like configuration can be obtained even for negative potentials. We show
that at the background level, this model can easily mimic the {\Lambda}CDM
model both with and without independent baryonic and radiation components.
