What's an efficient way to soundproof the space between my door frame and wall frame?
Noiseproofing, as you know, is best assisted by decoupling dense masses.

I am going to presume that your walls are framed thicker than standard 2x4 (as staggered 6 is a very common technique)

In this case, you have a pretty wide cap to fill in the shimmed jamb. 

I would shove fiberglass into the cavity, or spray in foam, making sure not to overfill!

Then I would lay in greenglue (or their gap-filling material), then I would lay in the backer rod. Finally, one more application of the gap-filling material before setting the casing. 

Here is a video of a door company demonstrating the install:
http://www.acousticalsurfaces.com/acoustic_doors/soundproof_doors.htm