avrdude 6.3 comes with a bit-banged GPIO driver that is suitable for use with this hat. All you need to do is add this to your avrdude.conf file:

programmer
    id = "pihat";
    desc = "Use the Linux sysfs interface to bitbang GPIO lines";
    type = "linuxgpio";
    reset = 25;
    miso = 9;
    mosi = 10;
    sck = 11;
;

This works, but there is a higher performance variant of this that actually uses the Pi SPI bus. To use it, you must first enable SPI in raspi-config. Once this is done, you'll see /dev/spi0.0 show up. Apply the patch (see below) to avrdude and then add this section to your avrdude.conf file:

programmer 
  id = "pihat"; 
  desc = "Use Linux SPI device in /dev/spidev*"; 
  type = "linuxspi"; 
  reset = 25; 
  baudrate=100000; 
; 

The patch to avrdude is here. This patch also adds support for using the TPI adapter for TPI programming.