  A geometrically frustrated lattice may host frustrated electron spin or
charge states that spawn exotic quantum phases. We show that a newly
synthesized metal-organic framework of Cu-Hexaaminobenzene [Cu$_3$(HAB)$_2$]
exhibits a multi spectra of unusual quantum phases long sought after in
condensed matter physics. On one hand, the Cu$^{2+}$ ions form an ideal $S$-1/2
antiferromagnetic kagome lattice. On the other hand, the conjugated-electrons
from the organic ligands give rise to completely dispersionless energy bands
around the Fermi level, reproducing a frustrated $\pi_x$-$\pi_y$ hopping model
on a honeycomb lattice. We propose to characterize the coexistence of
frustrated local spins and conjugated electrons through scanning tunneling
microscopy simulations. Most remarkably, their close energy proximity enables
one to tune the system between the two frustrated states by doping up to one
hole per HAB unit. Thus, Cu$_3$(HAB)$_2$ provides a unique exciting platform to
investigate the interplay of frustrated spins and electrons in one single
lattice, e.g. by gating experiments, which will undoubtedly raise interesting
theoretical questions leading to possible new condensed-matter phases.
