I'm currently writing an application which is doing something similar but I'm using Angular JS which has it's own views/partials which get loaded into the specified region in the page. The additional benefit is that if you make a change to a form field etc it is automatically updated between the model and the view.

There is no "clever" way, a simple ajax call to a controller method that returns your HTML which can be inserted back into the DOM is the most obvious way. As @ghodder mentioned, if this is an application wide requirement and will be needed a lot you are better combining Laravel with one of the many "one page" javascript libraries such as emberjs or angularjs where Laravel is merely an API.