How to edit/replace Parent functions.php function in Child Theme to add “Walker” class

Go to your functions.php in your child theme add this:

function childtheme_override_tesseract_output_menu( $cont, $contClass, $location, $depth ) {
//tesseract_output_menu function code modified as you wish here    
}

In your parent theme functions.php, wrap the tesseract_output_menu function like this:

if (function_exists('childtheme_override_tesseract_output_menu')) {

    /**
     * run the child override
     */
    function tesseract_output_menu() {
        childtheme_override_tesseract_output_menu();
    }

} else {
   /**
    * run the original parent function
    */
   function tesseract_output_menu( $cont, $contClass, $location, $depth ) {
     //original code untouched   
   }
}

this way you can override the parent theme function, this is the standard way, i am not sure why the parent theme function its not already wrapped like that, you can see in the themes that come with wordpress the same code for it, so the function can be replaced if ( ! function_exists( 'twentysixteen_setup' ) ).