Flutter web is amazing. I have the opportunity to build a side project with it last year, and I just love it. Compared to trying to build web applications with React or Vue, Flutter doesn't have to deal with the baggage of the JavaScript ecosystem.
Here's a server side Dart framework
Not sure how mature it is, but if I ever have a need to build another backend server I'll try it. I tend to prefer managed solutions now like Google's Firebase, which integrates so well with Flutter ( obviously because Google has a vested interest in you using their services)