(Update: As LanX also said in the meantime,) See the documentation for BUILDARGS in Moo:

use warnings;
use strict;

package MyDate {
    use Moo;
    use Types::Standard qw/ Int /;
    use Carp;
    use Date::Calc qw/Today/;
    use namespace::clean;
    has year  => ( is=>'ro', required=>1, isa=>Int );
    has month => ( is=>'ro', required=>1, isa=>Int );
    has day   => ( is=>'ro', required=>1, isa=>Int );
    around BUILDARGS => sub {
        my ( $orig, $class, %args ) = @_;
        my %newargs;
        # NOTE: This does not handle incorrect arguments!
        if ( keys %args ) {
            if ( keys %args == 3 ) {
                %newargs = %args;
            }
            elsif ( $args{date} ) {
                @newargs{qw/year month day/}
                    = $args{date}=~/^(\d{4})(\d\d)(\d\d)$/
                        or croak "bad date $args{date}";
            }
        }
        else {
            my ($year,$month,$day) = Today();
            %newargs = (year=>$year, month=>$month, day=>$day);
        }
        return $class->$orig(%newargs);
    };
    sub ymd {
        my $self = shift;
        return $self->year.'-'.$self->month.'-'.$self->day;
    }
}

print "1. ", MyDate->new()->ymd, "\n";
print "2. ", MyDate->new(year=>2019, month=>5, day=>14)->ymd, "\n";
print "3. ", MyDate->new(date=>20190512)->ymd, "\n";

__END__

1. 2019-5-16
2. 2019-5-14
3. 2019-05-12
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Update: To be clear, this is just a template (as requested), since it doesn't really do any checking of the constructor args. Personally, I also like to add use MooX::StrictConstructor; after use namespace::clean; to prevent typos.