Just because I have dismissed Xamarin as my cross platform solution does not mean that I am waking away from the Microsoft world. To the contrary I still build my backends in ASP.NET using C# . we use Entity Framework Code First to deliver database solutions that can be hosted in Azure, AWS or shared hosts like Winhost. The combination is great as we use Web Api to expose our server web methods to Flutter which has great HTTP support (see my blog post). Now there are lots of articles on Flutter and Dart so I am mostly zooming in on using .NET to deliver the backend services for my Flutter apps. .Net is a great server technology whether it is running on a shared host like Winhost or Microsoft's Azure and even AWS. .NET Core is a interesting new twist as it runs on non Windows platforms like Mac OS and Linux . Now it does not have all the functions of the original but is good enough to allow me to use my native Mac OS for .NET development. Additionally it is Open Source (this is the new Microsoft) .
I really believe that the combination of Flutter for portable UX and .NET for backend is powerful and one that your organization should be looking at. As such we would love to assist you on shortening your learning time and even helping with projects.
On my blog you will find a couple of posts on making secure calls (OAuth) to an ASP.NET Web Api backend. One is the ASP.NET part which uses Identity to provide the authentication. The second builds a simple login page in Flutter rhat uses these service.