  We design from first principles a new type of two-dimensional metal-organic
frameworks (MOFs) using phenalenyl-based ligands to exhibit a half-filled flat
band of the kagome lattice, which is one of the lattice family that shows
Lieb-Mielke-Tasaki's flat-band ferromagnetism. Among various MOFs, we find that
$\textit{trans}$-Au-THTAP(trihydroxytriaminophenalenyl) has such an ideal band
structure, where the Fermi energy is adjusted right at the flat band due to
unpaired electrons of radical phenalenyl. The spin-orbit coupling opens a band
gap giving a non-zero Chern number to the nearly flat band, as confirmed by the
presence of the edge states in first-principles calculations and by fitting to
the tight-binding model. This is a novel and realistic example of a system in
which a nearly flat band is both ferromagnetic $\textit{and}$ topologically
non-trivial.
