Place the joystick on top of the breakout board -- it only fits one way.
Flip the board over and solder all the connections on the bottom side.
Solder the header pins into the breakout board. It's easiest to do this if you have a solderless breadboard to hold everything in place.
Solder the header into a perma-proto breadboard, centering the joystick left to right but aligning it with the top edge, leaving a few pins open below the breakout board.
Solder a very long wire into the VCC, G, and Xout pins. (Photo shows a wire in each of the 5 pins, but you only need those three for this project). These wires will need to reach from the pocket to the Circuit Playground, so give yourself plenty of slack.
Put the hat on and put your hands in the pockets. Decide where you'd most like the joystick to be and make a small hole in the lining of the pocket for it to poke through.
Cut a larger hole in the edge of the pocket lining and slip the breadboard assembly inside. Poke the mechanism through, then snap the plastic joystick onto the whole assembly.
Stitch the breadboard in place using the mounting holes on either end.
Carefully cut a hole in the lining of the hat right at the crown of the head that's big enough to give yourself room to work.
Thread all the joystick wires up through the lining to the top of the head where the Circuit Playground is going to rest.
Temporarily attach the wires to the Circuit Playground with alligator clips, so you can test the servos in the next step.