I don't think you can take references to class methods like that.
The obvious solution (to me at least) is to make the package name the variable: 

sub _fill_lists {
    my ($thing, $thing_ref) = @_;

    my $class = "My::Class::$thing";                                 #
+ <==

    if (ref $giant_hash{$thing} eq 'ARRAY') {
        for (@{ $giant_hash{$thing} }) {
            push @{ $thing_ref }, $class->new( $_ );                 #
+ <==
        }
    } else {
        push @{ $thing_ref }, $class->new( $giant_hash{$thing} );    #
+ <==
    }

}

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In reply to Re: Need Help Refactoring Method Calls by rhesa
in thread Need Help Refactoring Method Calls by jffry

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