You want sysread instead of read. read will read until the specified number of bytes has arrived. Say your header is 500 bytes long and there's no body (e.g. a GET request), read will wait for another 524 bytes more to come even though more won't come. sysread will return as soon as some bytes are available.

my $buf = '';
my $header;
for (;;) {
   my $rv = sysread($sock, $buf, BLK_SIZE, length($buf));
   die("Can't read: $!\n") if !defined($rv);
   die("Can't read: Premature EOF\n") if !$rv;
   
   if ($buf =~ s/^(.*\r\n\r\n)//s) {
      $header = $1;
      last;
   }
}
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Use the same technique to read the body. Don't forget that some of the body may already be in $buf.