JSFiddle - React, Tailwind, and code Playground

HTML

<select id="signed_by">
    <option value="1">Test</option>
</select>

JavaScript

$(function() {
    var selectedOption = "19084";
    var select = $('#signed_by');

    options = select[0].options;

    $('option', select).remove();

    var obj = {
        "19081": "Becky Doe",
        "19082": "Aaron Smith",
        "19084": "Darren Hayes",
        "19083": "Adam Schwedler"        
    };
    
    $.each(obj, function(val, text) {
        options[options.length] = new Option(text, val);
    });
    
    // convert OPTIONs NodeList to an Array
    // - keep in mind that we're using the original OPTION objects
    var ary = (function(nl) {
        var a = [];
        for (var i = 0, len = nl.length; i < len; i++) {
          a.push(nl.item(i));
        }
        return a;
    })(options);
    // sort OPTIONs Array
    ary.sort(function(a, b) {
        return a.text < b.text ? -1 : a.text > b.text ? 1 : 0;    
    });
    // remove all OPTIONs from SELECT (don't worry, the original
    // OPTION objects are still referenced in "ary") ;-)
    options.length = 0;
    
    // (re)add re-ordered OPTIONs to SELECT
    select.html(ary);

    select.prepend($('<option />').text('-- Select Signed By --').val(''));

    select.val(selectedOption);
});