open my $up, "<", "$_[0]" or die "Something went wrong: ".$!; binmode 
+$up;
my $ck = Digest::MD5->new; $ck->addfile($up); close $up;
my $us = HTTP::Headers->new; $us->header(Content_Type => 'multipart/fo
+rm-data');
my $req = HTTP::Request->new('POST',"https://".$server."/file_bucket_f
+ile/md5/".$_[1]."/".$ck->hexdigest, $us, [ file => ["$_[0]"] ]);
print Dumper($req);
my $res = $ua->request($req);
die "Something went wrong: ".$res->status_line unless $res->is_success
+;
[download]

Something is wrong with the syntax though, and I can't understand

Nothing is wrong with the syntax, there are no syntax errors

get "Can't call method "request" on an undefined value ". $req is defined, and shown on the line above,

But you aren't calling any method request on variable $req. See diagnostics

You are right, I was staring at the undefined $ua (just a typo) and not seeing it, thank you.
Still, this doesn't seem to work in my test setup, i.e. apache as proxy in front of puppet. I get a "..timeout specified has expired: proxy: prefetch request body failed to 127.0.0.1:18140" in apache's log.
GETs do work, just PUTs fail. Config is

Listen 8140
<Proxy balancer://puppetmaster>
        BalancerMember http://127.0.0.1:18140
        BalancerMember http://127.0.0.1:18141
</Proxy>
<VirtualHost *:8140>
        SSLEngine On
        SSLCipherSuite SSLv2:-LOW:-EXPORT:RC4+RSA
        SSLCertificateFile /var/lib/puppet/ssl/certs/server.pem
        SSLCertificateKeyFile /var/lib/puppet/ssl/private_keys/server.
+pem
        SSLCertificateChainFile /var/lib/puppet/ssl/ca/ca_crt.pem
        SSLCACertificateFile /var/lib/puppet/ssl/ca/ca_crt.pem
        SSLCARevocationFile /var/lib/puppet/ssl/ca/ca_crl.pem
        SSLVerifyClient require
        SSLVerifyDepth 1
        SSLOptions +StdEnvVars
        RequestHeader set X-Client-DN %{SSL_CLIENT_S_DN}e
        RequestHeader set X-Client-Verify %{SSL_CLIENT_VERIFY}e
        <Location />
                SetHandler balancer-manager
                Order allow,deny
                Allow from all
        </Location>
        ProxyPass / balancer://puppetmaster/
        ProxyPassReverse / balancer://puppetmaster/
        ProxyPreserveHost On
        ErrorLog /mirror/log/balancer_error_log
        CustomLog /mirror/log/balancer_access_log "%h %l %u %t \"%r\" 
+%>s %b
\"%{SSL_CLIENT_S_DN}x\" \"%{User-Agent}i\""
</VirtualHost>
[download]

and the perl code is

{
    open my $up, "<", "$_[0]" or die "Something went wrong: ".$!; binm
+ode
$up;
    my $ck = Digest::MD5->new; $ck->addfile($up); close $up;
    my $req = HTTP::Request::StreamingUpload->new(
        PUT => "https://
".$server."/file_bucket_file/md5/".$ck->hexdigest,
        path => $_[0],
        headers => HTTP::Headers->new(
            'Content-Type' => 'application/binary',
            'Content-Length' => -s $_[0],
            'Accept' => 's',
        ),
    );
    my $res = $ua->request($req);
    die "Something went wrong: ".$res->status_line unless $res->is_suc
+cess;
    return $res->content;

}
[download]

SSL handshake does work, but the puppet process doesn't seem to receive any data (I'm tailing its debug log). Not sure if I am doing something wrong, or it is a problem in mod_proxy..