<?php $specialties = $wpdb->get_results("SELECT * FROM specialties ORDER by name ASC"); ?>
            <select class="form-control" name="specialty">
              <?php foreach($specialties as $specialty){ ?>
              <option value="<?php echo $specialty->id; ?>"><?php echo $specialty->name; } ?></option>
            </select>

$s = esc_sql($_GET['specialty']);

$sql = "SELECT id, name
FROM specialties
    WHERE";
if(!empty($s)) {
  $sql .= " specialty LIKE '%$s%'";
}