We study the half-filled Hubbard model on the triangular lattice with spin-dependent Kitaev-like hopping. Using the variational cluster approach, we identify five phases: a metallic phase, a non-coplanar chiral magnetic order, a $120^∘$ magnetic order, a nonmagnetic insulator (NMI) and an interacting Chern insulator (CI) with a nonzero Chern number. The transition from CI to NMI is characterized by the change of the charge gap from the indirect band gap to the direct Mott gap. The NMI phase is further suggested to be a gapless Mott insulator with a spinon Fermi surface or a fractionalized CI with spinon edge states depending on the strength of Kitaev-like hopping, based on the slave-rotor mean-field theory. Our work highlights the rising field that interesting phases emerge from the interplay of band topology and Mott physics.