Preheat the oven to 350 degrees . Using the large holes of a box grater, the teeth of a handheld slicer or the shredding attachment of a food processor, shred the potatoes; transfer to a large bowl. Add three-quarters of the melted butter and toss well. Season with 1 1/4 tsp. salt and toss to coat.
In a 9-inch nonstick skillet, heat the oil until rippling-hot. Remove the skillet from the heat and add the potatoes by the handful, squeezing out any excess liquid before adding; press flat to form a cake. Place the skillet back on the stove and cook over high heat until the bottom edges start to brown, about 8 minutes. If the cake seems dry, add a little more butter and loosen the edges by running a spatula gently around the sides to prevent sticking or burning. Remove the skillet from the heat and pour off any excess cooking fat (this will prevent splattering when you flip the cake). Invert a large plate onto the skillet and, using oven mitts, turn the cake out onto the plate. Add the remaining melted butter to the pan, slide in the cake browned side up and cook the second side until golden-brown, about 5 minutes. Transfer the pan to the oven and cook until the cake is crisp on the edges but still tender when pierced in the center with the tip of a knife, 10 to 12 minutes.
Drain off any excess cooking fat from the pan. Season the top of the potato cake with salt and turn it salted side down onto a large platter; season the second side with salt. Cut into 8 wedges and serve immediately.