Shred potatoes by hand or with the fine shred setting on the food processor.
Melt butter in a cast iron pan over medium-high heat. Press shredded potatoes, garlic, and thyme evenly into the pan and cook for 15 minutes until the bottom is completely browned. With a spatula, carefully create some space between the pan and the cake. Place a large plate over the top of the pan and flip cake onto the plate.
Add a bit more butter and slide cake back into the pan and brown on the opposite side.