In this talk I will discuss how the structural features of Kuznetsov components of Fano varieties are reflected in the geometry of their mirror Landau-Ginzburg models. I will focus on Fano varieties arising as cyclic covers of weighted projective spaces branched along Calabi-Yau or Fano hypersurfaces. I will discuss the case of the quartic double solid in detail. The talk is based on work in progress with Ilaria Di Dedda, Danil Kozevnikov, and Nick Sheridan.