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Two-dimensional helium-3 (³He) provides a simple model for the experimental investigation of the emergence of quantum complexity in a strongly correlated Fermi system. We have observed two-dimensional, two-band heavy-fermion behavior in bilayer films of ³He atoms when adsorbed on the surface of graphite preplated by a solid bilayer of ⁴He. Thermodynamic measurements on this system showed that the relevant control parameter is the total density of the ³He film. The ³He bilayer system can be driven toward a quantum critical point at which the effective mass appears to diverge, interband coupling vanishes, and a local-moment state appears. It opens a new testing ground for theories of quantum criticality in heavy-fermion materials.