Options for soldering a through hole PCB (e.g. Teensy) flush/flat on another PCB

1. Stick the headers through your main board from the bottom, with the plastic on the bottom.
2. Set your Teensy on top, with the headers going through its holes.
3. Solder the Teensy to the headers.
4. Using a small screwdriver, pry the plastic off the headers. You'll be surprised how easy it comes off.
5. Solder the headers to the main board.
6. Trim headers as necessary.

Have you considered simply surface-mounting the module onto your PCB? The header holes are big enough for solder to flow down and wet onto the main board. Or you could use some solder paste and reflow on a hot-plate or an oven.

1. A Kapton tape layer between the two boards to prevent accidental short circuits. (Must be Kapton to withstand soldering temperatures)

2. Just use resistor legs (or other sources of tinned wire) through each hole, soldered both sides, trimmed to length.

This isn't something you'll do in any quantity, or else it would be worth re-laying out the board (or cloning the Teensy with castellated pins to suit your purpose)