I bet you can, but this will probably require some nice WordPress coding skills.
This can be done by extending the themes (one bye one…hélas) or by witing a plugin that adds the link programmatically to all themes.
add to theme’s functions.php
(or add as a plugin to your /wp-content/mu-plugins/)
add_action( 'admin_bar_menu', 'toolbar_link_to_mypage', 999 );
function toolbar_link_to_mypage( $wp_admin_bar ) {
$args = array(
'id' => 'my_page',
'title' => 'My Page',
'href' => 'http://mysite.com/my-page/',
'meta' => array( 'class' => 'my-toolbar-page' )
);
$wp_admin_bar->add_node( $args );
}
https://codex.wordpress.org/Class_Reference/WP_Admin_Bar/add_node
http://codex.wordpress.org/Must_Use_Plugins