Building from source with BOARD=PICO_R1_3

Just a few hours later and some perseverance and I got it working =) . The method under Linux is:

make clean && BOARD=PICO BOOTLOADER=1 make

Then short BTN to BOOT0 under the board, indicated by a "<<" on the schematic circuit diagram.
Plug the device into the USB port while holding down the button and run the following command:

sudo dfu-util -a 0 -s 0x08000000 -D bootloader_espruino_2v08.5_pico_1r3.bin

If the command exits without displaying a progress bar, pull the board out of USB, plug it in again with the button held, and issue the above command again. Just a little finicky.

Once the bootloader is successfully flashed, unplug and remove the BTN / BOOT0 short, plug it back in with the button held. The red and green LEDs on the board should alternate lighting up.

Now issue the following command:

make clean && BOARD=PICO make serialflash

The firmware will build and the STM32F401 will automatically decide which address to place the remainder of the firmware in its flash memory.

Unplug the board, plug it back in (no need to short or hold any button down this time)

picocom --baud 9600 --flow n /dev/ttyACM0

Then press Ctrl+C once.

This will spawn a terminal with which to send Javascript commands and functions to the espruino.