As it seems that your mail server is exim, you can send the messages to it through a pipe using "batch SMTP":

open my $exim, "| exim -bS" or die;

for (...) {
  print $exim <<'MSG';
MAIL FROM: foo@bar.com
RCPT TO: doz@bar.com
DATA
Subject: Your Invoice
To: You
From: Me

You have to pay 1,000,000 dollars
.
MSG

}
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If you want to parallelize, fork some processes and divide the task between them:

my $workers = 4;
for my $ix (0..$workers-1) {
  unless (fork) {
    open my $fh, '<', $address_list_file_name or die;
    open my $exim, "| exim -bS" or die;
    while (<>) {
      next unless (($. % $workers) == $ix);
      send_invoice($exim, $_);
    }
    exit 0;
  }
}
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