JSFiddle - React, Tailwind, and code Playground

HTML

<script src="http://ajax.googleapis.com/ajax/libs/jquery/1.4.1/jquery.js"></script>
<script src="http://ajax.googleapis.com/ajax/libs/jqueryui/1.7.2/jquery-ui.min.js"></script>
<link rel="stylesheet" href="http://ajax.googleapis.com/ajax/libs/jqueryui/1.7.2/themes/base/jquery-ui.css">
    <label for="startDate">Date :</label>
    <input name="startDate" id="startDate" class="date-picker" />

CSS

.ui-datepicker-calendar {
    display: none;
    }

JavaScript

$(function() {
     $('.date-picker').datepicker(
                    {
                        dateFormat: "mm/yy",
                        changeMonth: true,
                        changeYear: true,
                        showButtonPanel: true,
                        onClose: function(dateText, inst) {


                            function isDonePressed(){
                                return ($('#ui-datepicker-div').html().indexOf('ui-datepicker-close ui-state-default ui-priority-primary ui-corner-all ui-state-hover') > -1);
                            }

                            if (isDonePressed()){
                                var month = $("#ui-datepicker-div .ui-datepicker-month :selected").val();
                                var year = $("#ui-datepicker-div .ui-datepicker-year :selected").val();
                                $(this).datepicker('setDate', new Date(year, month, 1)).trigger('change');
                                
                                 $('.date-picker').focusout()//Added to remove focus from datepicker input box on selecting date
                            }
                        },
                        beforeShow : function(input, inst) {

                            inst.dpDiv.addClass('month_year_datepicker')

                            if ((datestr = $(this).val()).length > 0) {
                                year = datestr.substring(datestr.length-4, datestr.length);
                                month = datestr.substring(0, 2);
                                $(this).datepicker('option', 'defaultDate', new Date(year, month-1, 1));
                                $(this).datepicker('setDate', new Date(year, month-1, 1));
                                $(".ui-datepicker-calendar").hide();
                            }
                        }
                    })
});