We propose a simple scheme for realizing large Chern numbers in an optical Floquet quasicrystal with cold atoms. Notably, quasicrystals host rich topological spectra and and topological gaps can be further opened largely by Floquet engineering. The corresponding Chern numbers are probed via particle transport in Thouless pumping, revealing a direct connection between pumping dynamics and the underlying topological spectra. Our work provides a simple and controllable route to realizing large Chern numbers, opening new avenues for exploring correlated topological states. For more details, see our preprint arXiv:2606.02489.