WP Cron job every 1st and 15th of the month

wp_cron operates on intervals and there is no interval that will hit exactly the first day and the 15th of every month.

You could run your wp-cron job every day and check the date, similar to this but with a simple callback like:

cron_callback_wpse_113675() {
  $date = date('d');
  if ('01' == $date || '15' == $date) {
    // run your function
  }
}

Or, use wp_schedule_single_event with a more complicated callback. Something like this:

function cron_callback_v2_wpse_113675() {
  $date = date('d');
  if (!wp_next_scheduled(cron_callback_v2_wpse_113675)) {
    $date = ($date < 15) ? '15' : '01';
  }
  if ('01' == $date || '15' == $date) {
    // run your function
    switch ($date) {
      case '01' :
        wp_schedule_single_event( strtotime('+14 days',strtotime('first day of')), 'cron_callback_v2_wpse_113675' );
      break;
      case '15' :
        wp_schedule_single_event( strtotime('first day of'), 'cron_callback_v2_wpse_113675' );
      break;
    }
  }
}

Barely Completely untested. Possibly buggy. Caveat emptor. No refunds.