Hmm, by default LWP::UserAgent should be sending a HTTP/1.1 request. LWP::UserAgent contains

if ($ENV{PERL_LWP_USE_HTTP_10}) {
    require LWP::Protocol::http10;
    LWP::Protocol::implementor('http', 'LWP::Protocol::http10');
    eval {
        require LWP::Protocol::https10;
        LWP::Protocol::implementor('https', 'LWP::Protocol::https10');
    };
}
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use strict;
use warnings;
use LWP;
use LWP::Debug '+';
use Data::Dumper;

my $ua = LWP::UserAgent->new();
$ua->get(shift || 'http://localhost');

# because LWP I<use>s it at runtime
use LWP::Protocol::http;
package LWP::Protocol::http::Socket;
sub format_request {
    my $self = shift;
    my $ret = $self->SUPER::format_request(@_);
        LWP::Debug::trace($ret);
    return $ret;
}
__END__
LWP::UserAgent::new: ()
LWP::UserAgent::request: ()
LWP::UserAgent::send_request: GET http://localhost
LWP::UserAgent::_need_proxy: Not proxied
LWP::Protocol::http::request: ()
LWP::Protocol::http::Socket::format_request: GET / HTTP/1.1
TE: deflate,gzip;q=0.3
Connection: TE, close
Host: localhost
User-Agent: libwww-perl/5.75

LWP::Protocol::collect: read 640 bytes
LWP::Protocol::collect: read 854 bytes
LWP::UserAgent::request: Simple response: OK
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