Rinse the farro and place it into a saucepan with the water and bring to a boil. Reduce the heat and allow to cook at a low boil, uncovered, for about 15 minutes. Drain off any excess water and place the farro into a bowl or Kitchen Aid mixer.
Bring a medium-sized pot of water to a boil. Add in the cubed sweet potato and cook until tender, about 12 minutes. Drain the sweet potatoes very well, mash, and measure out 1 cup, then add it with the farro. Any leftover sweet potatoes can be eaten later or used to make croquettes or gyozas.
Add in the rest of the burger ingredients and knead with your hands or in a Kitchen aid mixer for 3-4 minutes, to allow the gluten to develop.
Shape the mixture into 6 equal pieces. Roll into spheres, then flatten them out between your palms a bit. You can either grill them immediately or store them in the refrigerator until you’re ready to grill.
To make the pickles
Bring the water and rice vinegar to a boil. Add in the salt and sugar and stir until dissolved. Add in the sliced cucumbers and stir. Sprinkle the seeds over the top, cover with a lid and remove from the heat. Once it’s cool enough, transfer to a glass container and allow to chill and marinate in the refrigerator for a few hours up to a couple of days before eating (they taste best after a few days). The pickles will last about a week in the refrigerator.