use DateTime qw( );

my $doy  = ...;  # 1-366

my $year = DateTime
           ->today( time_zone => 'local' )
           ->year();

my $dow  = DateTime
           ->from_day_of_year(
              year        => $year,
              day_of_year => $doy,
           )
          #->day_of_week();   # 1-7 = M-Su, or
           ->day_name();      # Name of day according to locale.