ArcGIS JSAPI: Query Distinct, the old fashion way.

gistfile1.js

require([..., "esri/tasks/query", "esri/tasks/QueryTask", "dojo/_base/array",...],
  function (..., Query, QueryTask, arrayUtils,...) {
  ...
  // urls faked to protect the innocent kitties
  var qTask = new QueryTask("http://www.cat-hurders.com:6080/arcgis/rest/services/CatSightings/MapServer/"),
    query = new Query();
  // set up the parameters to query for the breeds
  query.where = "1=1";
  query.outFields = ["breed"];
  query.returnGeometry = false;
  // execute the query
  qTask.execute(query).then(function (featureSet) {
    var features, i;
    // collect the list of breeds
    features = arrayUtils.map(featureSet.features, function (feature) {
      return feature.attributes.breed;
    });
    // sort the list
    features.sort();
    // pick through each item in the list, and if it matches the item before, remove it.
    for (i = features.length - 1; i > 0; i--) {
      if (features[i] === features[i-1]) {
        features.splice(i, 1);
      }
    }
    // do something with the list.
    ...
  });
  ...
});