Optical metasurfaces with high-Q chiral resonances can boost light-matter interaction for various applications of chiral response for ultrathin, active, and nonlinear metadevices. Usually, such metasurfaces require sophisticated depth-resolved nanofabrication to realize subwavelength stereo-nanostructures, posing overwhelming challenges, especially in the short-wavelength range. Here, we suggest a novel planarized design of chiral metasurfaces supporting bound states in the continuum (BICs), and demonstrate experimentally chiroptical responses with record-high Q-factors (Q=390) and near-perfect circular dichroism (CD=0.93) at optical frequencies. The symmetry-reduced meta-atoms are highly birefringent and support winding elliptical eigen-polarizations with opposite helicity surrounding the BIC polarization singularity, providing a convenient way for achieving maximal planar chirality tuned by either broken in-plane symmetry or changing illumination direction. Such sharply resonant chirality realized in planar metasurfaces promises various practical applications in classical and quantum optics including chiral sensing, enantiomer selection, and chiral quantum emitters.