WordPress and Ajax - reload shortcode content

I would wirte a comment, but my current reputation only lets me to write an answer. Here id my solution to reprint the content of the AJAX output.

On PHP, add a container div with ID:

function contentDemoConnecteurs(){
    $html = '<div id="projectSelector">';

    $html .= 'Choix du projet : ';        
    $html .= '<select id="projectChosen" name="project">';
    foreach($this->projects as $p) {
        $selected = ($p->id == $this->project_id) ? 'selected="selected"' : '';
        $html .= '<option value="' . $p->name .'" '. $selected . ' >' . $p->name . '</option>';
    }
    $html .= '</select><br>';

    $html .= 'Choix de la version : ';
    if (isset ($this->versions)){
        $html .= '<select id="versionChosen" name="version">';
        foreach($this->versions as $v) {
            $selected = ($v->id == $this->version_id) ? 'selected="selected"' : '';
            $html .= '<option value="' . $v->id .'" '. $selected . ' >' . $v->id . '</option>';
        }
        $html .= '</select>';
    }
    $html .= '</div>';
    return $html;
}

On JQuery:

jQuery(document).ready(function($) {
    $('#projectChosen').on('change', function () {
        jQuery.ajax({
            type: "POST",
            url: ajax_object.ajaxurl,
            data: {
                'action': 'setProjectChosen',
                'demo_projet_name': $('#projectChosen option:selected').val()
            },
            success: function (output) {
                $( "div#projectSelector" ).replaceWith(output);
            },
            error: function(errorThrown){
                console.log(errorThrown);
            }
        });
    } );
} );

I hope that's what you need.

I finally made it, I got confused about what I wanted to do...

<?php 
/**
 *   Plugin Name: demoConnecteurs
 *   Description: Plugin de démo des connecteurs Jenkins et Mantis
**/
require_once(file_with_external_fonctions.php);

$inst_demoConnecteurs = new demoConnecteurs();
if (isset($inst_demoConnecteurs)){
}

class demoConnecteurs{   
    private $projects;
    private $versions;

    private $project_id;
    private $project_name;
    private $version_id;


    function __construct(){
        $this->setProjects();

        $this->initAjaxActions();

        add_action('admin_enqueue_scripts', array($this,'demo_scripts'));
        add_action('wp_enqueue_scripts', array($this,'demo_scripts'));


        $this->init();
    }

    function initAjaxActions(){
        add_action('wp_ajax_setProjectChosen', array($this,'setProjectChosen'));
        add_action('wp_ajax_nopriv_setProjectChosen', array($this,'setProjectChosen'));
    }

    function demo_scripts(){
        wp_register_script( 'ajaxHandle', plugins_url() . '/DemoConnecteurs/buttons_ajax.js');
        wp_localize_script( 'ajaxHandle', 'ajax_object', array( 'ajaxurl' => admin_url( 'admin-ajax.php' ) ) );

        wp_enqueue_script( 'ajaxHandle');
    }

    function init(){
        add_shortcode( 'demoConnecteurs', array($this,'demoConnecteurs_shortcode') );
    }

    function demoConnecteurs_shortcode () {
        return $this->contentDemoConnecteurs();
    }

    public function setProjects(){
        $this->projects = getProjects();
    }

    public function setProjectChosen(){
        if (isset ($_POST['demo_projet_name']))$this->project_name = $_POST['demo_projet_name'];
        $this->project_id = getProjectIdFromName($this->mantisClient,  $_SESSION['username'], $_SESSION['password'], $this->project_name);

        $this->setVersions();
        echo $this->contentDemoConnecteurs();
        wp_die();
    }

    public function setVersions(){
        $this->versions = getVersionsOfProjectChosen($this->project_id);
    }

    function contentDemoConnecteurs(){
        $html = '<div id="contentDemoConnecteurs">';

        $html .= 'Choix du projet : ';        
        $html .= '<select id="projectChosen" name="project">';
        foreach($this->projects as $p) {
            $selected = ($p->id == $this->project_id) ? 'selected="selected"' : '';
            $html .= '<option value="' . $p->name .'" '. $selected . ' >' . $p->name . '</option>';
        }
        $html .= '</select><br>';

        $html .= 'Choix de la version : ';
        if (isset ($this->versions)){
            $html .= '<select id="versionChosen" name="version">';
            foreach($this->versions as $v) {
                $selected = ($v->id == $this->version_id) ? 'selected="selected"' : '';
                $html .= '<option value="' . $v->id .'" '. $selected . ' >' . $v->id . '</option>';
            }
            $html .= '</select>';
        }
        $html .= '</div>';
        return $html;
    }
}

jQuery(document).ready(function($) {

    $('#projectChosen').on('change', function () {
        jQuery.ajax({
            type: "POST",
            url: ajax_object.ajaxurl,
            data: {
                'action': 'setProjectChosen',
                'demo_projet_name': $('#projectChosen option:selected').val()
            },
            success: function (output) {
                $('#contentDemoConnecteurs').replaceWith(output);
            },
            error: function(errorThrown){
                console.log(errorThrown);
            }
        });
    } );
} );