Options for soldering a through hole PCB (e.g. Teensy) flush/flat on another PCB
Stick the headers through your main board from the bottom, with the plastic on the bottom. 
Set your Teensy on top, with the headers going through its holes. 
Solder the Teensy to the headers. 
Using a small screwdriver, pry the plastic off the headers. You'll be surprised how easy it comes off.
Solder the headers to the main board. 
Trim headers as necessary.