JSFiddle - React, Tailwind, and code Playground

HTML

<select name="event_month" id="event_month">
    <option value="1">Jan</option>
    <option value="2">Feb</option>
    <option value="3">Mar</option>
    <option value="4">Apr</option>
    <option value="5">May</option>
    <option value="6">Jun</option>
    <option value="7">Jul</option>
    <option value="8">Aug</option>
    <option value="9">Sep</option>
    <option value="10">Oct</option>
    <option value="11">Nov</option>
    <option value="12">Dec</option>
</select>

<select name="event_day" id="event_day">
    <!-- this will be filled by Javascript once the month is selected -->
</select>

JavaScript

$('#event_month').bind('change', function() {

    // For this test we'll stick with 2012
    var selected_year = '2012';
    
    // This is the month the user just selected
    var selected_month = $('#event_month').val();

    // Find out how many days in selected month
    var days_in_month = new Date(selected_year, selected_month, 0).getDate();

    // This is our days <select> element, which we'll fill with <option> for the month
    var select_element = $('#event_day');

    // Just an array with the days in our desired language
    var days_of_week = ["Sun","Mon","Tue","Wed","Thu","Fri","Sat","Sun"];

    // Clear the element of the last selected month's values
    select_element.empty();

    // For each day in this month...
    for (var day=1; day<=days_in_month; day++) {
        // ...get the day's name...
        var day_name = days_of_week[new Date(selected_year, selected_month-1, day).getDay()];
        // ...add the name and number in an <option> element
        select_object.append('<option value="'+day+'">'+day_name+' '+day+'</option>');
    }

});