In the spirit of TIMTOWTDI, here's a way without BUILDARGS. All the attributes are lazy and depend on each other. I used Time::Piece instead of Date::Calc::Today, but that's not relevant to the question.

#! /usr/bin/perl
{   package MyDate;
    use Moo;
    use Time::Piece;
    use namespace::clean;

    has year  => (is => 'lazy');
    has month => (is => 'lazy');
    has day   => (is => 'lazy');

    has date  => (is => 'lazy', predicate => 'has_date');

    sub _from_date {
        my ($self, $what, $pos, $length) = @_;
        return substr $self->date, $pos, $length if $self->has_date;
        return localtime->$what
    }

    sub _build_year  { $_[0]->_from_date(year => 0, 4) }
    sub _build_month { $_[0]->_from_date(mon  => 4, 2) }
    sub _build_day   { $_[0]->_from_date(mday => 6, 2) }

    my %format = ( year => '%04d' );
    sub _build_date  {
        my ($self) = @_;
        join "", map sprintf($format{$_} || '%02d', $self->$_),
                     qw( year month day )
    }
}

use Test::Spec;

describe MyDate => sub {

    it 'initializes from now' => sub {
        Time::Piece->expects('year')->returns(2019);
        Time::Piece->expects('mon')->returns(5);
        Time::Piece->expects('mday')->returns(16);

        my $md = 'MyDate'->new();
        is $md->date, '20190516';
    };

    it 'initializes from date' => sub {
        my $md = 'MyDate'->new(date => 20190201);
        is_deeply [ $md->year, $md->month, $md->day ], [2019, '02', '0
+1'];
    };

    it 'initializes from ymd' => sub {
        my $md = 'MyDate'->new(year => 2019, month => 11, day => 3);
        is $md->date, '20191103';
    };
};

runtests();
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