Thank you Benjamin, your suggestion was correct. Here is the final dart code.
void drawCzmlStream() {
_viewer.callMethod('extend', [context['Cesium']['viewerDynamicObjectMixin']]);
_viewer['dataSources'].callMethod('removeAll');
var czmlDataSource = new JsObject(context['Cesium']['CzmlDataSource']);
var loaded = czmlDataSource.callMethod('loadUrl', ['../czml/Vehicle.czml']);
loaded.callMethod('then', [(_) => displaySource(czmlDataSource)]);
print("finished");
}
void displaySource(dataSource) {
_viewer['dataSources'].callMethod('add', [dataSource]);
print("finished displaying");
}