A JavaFX application exists and the application is starting from the Main.class file which extends Application:
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You can implement your own controller factory so you would be able of creating your own controllers context.
I have used spring with javaFX, and I have implemented my own controllers factory using spring, in that way you can inject one controller in other (you shouldn't need it as those connections should be on the model)
If you want to try spring with Java fx: How to wire multiple fxml controllers using spring dependency-injection?