The quantum anomalous Hall effect carries dissipationless chiral edge currents without an external magnetic field, yet raising its operating temperature and controlling the number of edge channels remain difficult. Using first-principles calculations, we identify the two-dimensional metal-organic kagome ferromagnet Eu₂(C₆H₄)₃ as an intrinsic Chern insulator whose topological gap can be enlarged by mechanical strain. The monolayer has a spin-orbit-coupling-induced gap of 72.7 meV that widens to 124.7 meV under -8% biaxial strain, realizing a quantum anomalous Hall phase with Chern number C=-1 and a single chiral edge state. The local moments come from the half-filled Eu 4f⁷ shell, while the gap-opening spin-orbit coupling is carried by Eu 5d states hybridized into the carbon kagome bands. The AB-stacked bilayer couples ferromagnetically and accumulates the per-layer Chern numbers, giving C=-2 with two co-propagating chiral channels. An out-of-plane electric field then drives the bilayer through a sequence of topological transitions among C=-2, -3, and -1, switching the number of edge channels. Together, strain, stacking, and gating give three distinct handles on the gap and the Chern number within a single stoichiometric material.