From Proof of Concept on BeagleBone to Prototype
I might understood it wrong, but if you already have a project (OS + Code + any external components) running on a BBB and wish to create your own neat board to run the same project you need to get started with the BBB documentation. Have a look on the schematics and BOM first: https://github.com/beagleboard/BeagleBone-Black.

BBB is an open-source project and you should be able to find all details how to build a clone and add/remove pieces of it to build a customised project.

Some observations


It will be close to impossible to have all Beagle Bone Black components running on a breadboard. You'll need to go directly to a proper multilayer PCB. You can make it a big PCB and add lots of testing points and additional pads to help you debugging it.
If that's your first PCB design project you might struggle as those little boards are very dense, using tiny components, including BGAs which will require professional assembly.
The cost for producing a similar board in small quantities (let's say under 10K) might end up being much over the cost of using the BBB itself. Building a "cape/shield" only and a nice enclosure to fit both boards together could be a better/cheaper solution.


If you really need to have your own board for any other reason, have a look on the RPI Customisation service from Element14: http://www.element14.com/community/docs/DOC-76955/l/raspberry-pi-customization-service.

Have fun!
Cheers