Peel the potatoes and grate on the large holes of a box grater. Transfer to a clean dish towel and squeeze out as much liquid as possible. Place the squeezed potatoes in a medium bowl and mix in the chopped thyme, 1 teaspoon kosher salt, 1/4 teaspoon black pepper and chipotle. Toss well to combine.
Heat 2 tablespoons of the oil in a medium nonstick skillet over medium heat until the oil is very hot but not smoking. Add the potato mixture and press down firmly using a spatula. Cook until browned and crispy on the bottom, 8 to 9 minutes, shaking the pan occasionally so it does not stick. The cake should move around as one piece when the skillet is shaken. Slide the cake out of the skillet onto a plate and add the remaining oil to the skillet. Flip the cake onto a second plate and slide back into the skillet, crispy side up. Continue to cook until the bottom side of the cake is browned and crispy and the cake is cooked through, another 8 to 9 minutes, lowering the heat if the cake is browning too quickly.
Slide onto a wire rack and cool until still warm but not hot. Cut into 6 wedges, lightly season with salt and fresh thyme leaves and serve.