You was right. But you still don't need to use POST. This is work too:

#!/usr/bin/perl
use strict;
use warnings;
use WWW::Mechanize;
my $agent = WWW::Mechanize->new();

$agent->get('http://www.usgbc.org/myUSGBC/Members/MembersDirectory.asp
+x?PageID=2011&CMSPageID=140');
$agent->form_name('frmMemberDirectory');
$agent->field('ddlCountry', 'US');
$agent->click('btnSearch');

$agent->submit_form(
    
    form_name => "frmMemberDirectory",
    fields => {
        
        '__EVENTTARGET' => 'dgMembers$ctl01$ctl01',
        '__EVENTARGUMENT' => "",
    },
);

open my $test, ">", "test.htm" or die $!;
print $test $agent->content();
close $test;
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