foobar.js

$(document).ready(function () {
    $.ajax({ type: "HEAD",      url: '/foobar', dataType: "html", }).done(function (data) { console.log(data) }); // NOK, gives 404
    $.ajax({ type: "LINK",      url: '/foobar', dataType: "html", }).done(function (data) { console.log(data) }); // OK
    $.ajax({ type: "UNLINK",    url: '/foobar', dataType: "html", }).done(function (data) { console.log(data) }); // OK
    $.ajax({ type: "FLUBBER",   url: '/foobar', dataType: "html", }).done(function (data) { console.log(data) }); // OK
    $.ajax({ type: "FOOBAR",    url: '/foobar', dataType: "html", }).done(function (data) { console.log(data) }); // OK
});

Foobar.pm

package Foobar;
use Mojo::Base 'Mojolicious';
use feature ':5.24';
use experimental qw(signatures refaliasing);
no warnings qw(experimental);

sub startup($app) {
    my $r = $app->routes;

    # neither of these work
    $r->any(['HEAD']    => '/foobar')->to(cb => sub { warn; shift->render(json => '"HEAD"' ) });
    $r->_generate_route(HEAD => '/foobar')->to(cb => sub { warn; shift->render(json => '"HEAD"' ) });

    # but these do work
    $r->any(['LINK']    => '/foobar')->to(cb => sub { warn; shift->render(json => '"LINK"' ) });
    $r->any(['UNLINK']  => '/foobar')->to(cb => sub { warn; shift->render(json => '"UNLINK"' ) });
    $r->any(['FLUBBER'] => '/foobar')->to(cb => sub { warn; shift->render(json => '"FLUBBER"' ) });
    $r->_generate_route(FOOBAR => '/foobar')->to(cb => sub { warn; shift->render(json => '"FOOBAR"' ) });
}

1;