Start by prepping your potatoes. Wrap the grated potatoes in a clean dish towel. Twist the towel tightly, wringing out as much moisture from the potatoes as humanly possible. (I find it’s easiest to do this in 2-3 batches.) Place the dried potatoes in a mixing bowl. Season with a generous amount of salt and fresh ground pepper. Briefly set aside.
Heat the butter in a 10-inch skillet over medium heat. (I recommend a well-seasoned cast iron or non-stick skillet.) When melted add the leeks and thyme, and season salt and fresh ground pepper. Cook the leeks for 8-10 minutes, stirring regularly, until very soft. (If the leeks start to brown, reduce the heat to medium-low and carry on.) Transfer the leeks to a bowl. Wipe out the skillet and return it to the stove.
Heat 1½ tablespoons of oil over medium-high until shimmering. (To check if the oil is hot enough, add a strand of potato to the skillet. If it sizzles enthusiastically, you’re good to go.) Add half of the potatoes to the pan in an even layer. Add the leeks in an even layer and then the goat cheese. Top with the remaining potatoes in an even layer.
Cook for 3 minutes, then lower the heat to medium-low. (You should still hear the potatoes sizzling.) Cook for another 13-15 minutes until the underside of the potatoes is golden brown and crisp.
Time to flip the rosti. To do this, place a large plate over the skillet and carefully turn the skillet over, flipping the rosti onto the plate.
Heat the remaining 1½ tablespoons of oil in the skillet. When hot, carefully slide the rosti from the plate back into the skillet. Cook for another 10-12 minutes until the underside is browned and the center is tender. Remove the rosti from the pan and allow it to cool slightly.
Slice the rosti into quarters and serve warm. Top each serving with a fried egg if you like