JSFiddle - React, Tailwind, and code Playground

HTML

<link rel="stylesheet" href="http://ajax.googleapis.com/ajax/libs/jqueryui/1.8.10/themes/dark-hive/jquery-ui.css">
Normal: <input type="text" class="DatePicker">
<p>No days: <input type="text" class="DatePicker noDays"></p>

CSS

div.noDays table {
    display:none;
}

JavaScript

$('.DatePicker').datepicker({
    showButtonPanel: true,
    changeMonth: true,
    changeYear: true,
    dateFormat:'MM yy',
    beforeShow: function(input, inst) {
        if ($(input).hasClass('noDays')) {
            $('#ui-datepicker-div').addClass('noDays');
        } else {
            $('#ui-datepicker-div').removeClass('noDays');
            $(this).datepicker('option', 'dateFormat', 'yy-mm-dd');
        }
    },
    onClose: function(dateText, inst) {
        if ($('#ui-datepicker-div').hasClass('noDays')) {
            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));
        }
    }
});