The recent discovery of fractional quantum anomalous Hall (FQAH) insulators in two-dimensional van der Waals materials opens the door to an amazing variety of strongly correlated electronic phases and phase transitions beyond the conventional quantum Hall setting at large magnetic fields. In this talk, I describe several novel compressible phases that can be accessed by doping incompressible FQAH states realized in a fractionally filled Chern band. Unlike in the lowest Landau level, where single-electron motion is confined into cyclotron orbits, electrons in a Chern band occupy Bloch states with well-defined crystal momenta. At a non-zero doping density, this enables the formation of itinerant states of the doped anyons just beyond the FQAH plateau region. Near the Jain state at 2/3 filling, we find three types of compressible phases, depending on which anyon has the lowest excitation gap. These include a topological superconductor with chiral neutral fermion edge modes as well as a more exotic Pair Density Wave (PDW) superconductor with coexisting non-Abelian topological order. In addition to superconductors, we also find a CDW metal that arises from pairing instabilities of a non-Fermi liquid that can emerge near filling 2/3 at intermediate temperatures. Though inspired by the theory of anyon superconductivity, our construction is qualitatively different, primarily due to filling constraints associated with the lattice translation symmetry. If time permits, I will also discuss the unusual transport signatures associated with each phase and analyze the crossover between different finite-temperature regimes.