For the onions: Put the red onions in a medium bowl. Combine the lime juice, vinegar, sugar and salt in a small saucepan. Bring to a boil and cook until the sugar and salt are dissolved. Remove from the heat and let sit to cool slightly, 5 minutes, then pour over the onions. Cover and refrigerate for at least 1 hour and up to 24 hours.
For the hash browns: Mix together the creme fraiche and green onions in a small bowl, and season with salt and pepper. Cover and refrigerate until ready to serve.
Grate the potatoes using the large holes on a cheese grater. Heat half of the clarified butter in a 12-inch seasoned cast-iron pan over medium-low heat. Add the garlic and cook until it is lightly golden brown on both sides and infuses the oil with the flavor of the garlic, then remove and discard.
Toss the potatoes with the poppy seeds, salt, sesame seeds and some pepper. Add the potatoes to the pan and stir to coat with butter. Using a spatula, gently press the potatoes into a flat round, slightly smaller than the pan. Cook, shaking the pan occasionally, until the edges are golden and crisp, about 10 minutes.
Working carefully with some large, flat spatulas, flip the potato cake. Pour the remaining butter around the edges of the potato cake and continue cooking until potatoes are crisp on the bottom, about 10 minutes. Transfer to a cutting board. Let cool for a few minutes. Spread the top with the prepared creme fraiche and scatter the pickled onions over the top. Garnish with parsley leaves and cut into wedges to serve.