  We propose two conjectures on a moduli theoretic approach to constructing
Lagrangian subvarieties of hyperk\"ahler varieties arising from the Kuznetsov
components of cubic fourfolds or Gushel--Mukai fourfolds. Then we verify the
conjectures in several cases, recovering classical examples. As a corollary, we
confirm a conjecture of O'Grady in several instances on the existence of
Lagrangian covering families for hyperk\"ahler varieties.
