Quasi-bound states in the continuum (QBICs) are Fano resonant states with long optical lifetimes controlled by symmetry-breaking perturbations. While conventional Fano responses are limited to linear polarizations and do not support tailored phase control, here we introduce QBICs born of chiral perturbations that encode arbitrary elliptical polarization states and enable geometric phase engineering. We thereby design metasurfaces with ultrasharp spectral features that shape the impinging wave front with near-unity efficiency. Our findings extend Fano resonances beyond their conventional limits, opening opportunities for nanophotonics, classical and quantum optics, and acoustics.