View exposed filter with month and remove year
Using CSS and jQuery you can achieve your requirement.

Step 1: Using jQuery, on change() event of month select list assign current year to year select list.

$(".date-month").change(function(){
    //write code to set current year to year select list.
});


Step 2: Now using CSS, hide year select list.

.date-year {
    display: none;
}



  Note: You can also achieve the same functionality using Drupal custom hooks.