An obvious approach is to convert times to seconds, find the midpoint, adjust and return, like so:

use 5.16.2;
use POSIX 'strftime';

my @test = (['11:00', '13:00'], ['23:00', '01:00'], ['10:30', '13:00']
+, ['19:40', '01:20']);

foreach my $tst (@test) {
    print "Testing '$tst->[0]', '$tst->[1]': ";
    say getmid($tst->[0], $tst->[1]);
}

sub getmid($$) {
    my ($start, $end) = @_;

    $start =~ /^(\d{2}):(\d{2})$/ or die "invalid start date provided:
+ '$start', must be in format hh:mm\n";
    my $startseconds = $1 * 3600 + $2 *60;

    $end =~ /^(\d{2}):(\d{2})$/ or die "invalid end date provided: '$e
+nd', must be in format hh:mm\n";
    my $endseconds = $1 * 3600 + $2 * 60;

    my $range;
    if ($endseconds < $startseconds) { $range = 86400 - $startseconds 
++ $endseconds }
    else                             { $range = $endseconds - $startse
+conds }

    my $average = $startseconds + int($range / 2);

    return strftime("%H:%M", gmtime($average));
}
[download]