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We study the resonant enhancement of linear and nonlinear chiroptical effects in asymmetric silicon metasurfaces supporting multipolar Mie resonances and quasi-bound states in the continuum (quasi-BICs). We demonstrate theoretically and observe in experiment the pronounced linear circular dichroism at the quasi-BIC resonances. We further find that both local field enhancement and third-harmonic signal are large for Mie resonances and some quasi-BIC modes. We explain the selectivity of the nonlinear enhancement by employing the concept of critical coupling being more favorable for the modes with moderately large radiative quality factors ($Q$ factors). We demonstrate experimentally strong nonlinear chiroptical response associated with high efficiency of the third-harmonic generation and large nonlinear circular dichroism varying from $+0.918\pm0.049$ to $-0.771\pm0.004$ for the samples with different asymmetries. We believe our results suggest a general strategy for engineering nonlinear chiroptical response in dielectric resonant metasurfaces.