Clarify the butter: Melt the butter in a small saucepan and bring it to a gentle simmer.
Shut off the heat and allow it to sit a minute on the stove. The milk solids should start to sink to the bottom. Slowly pour the butter into a bowl, keeping as much of the white milky liquid as possible in the saucepan. Discard the milk solids. These are prone to burning and by clarifying the butter, the potato cake will be less likely to overbrown. Keep the butter warm on the stove.
Prepare the potatoes: Using a mandoline slicer or a sharp knife, cut the potatoes into thin 1/8-inch-thick slices. Transfer them to a bowl, sprinkle with a pinch of salt and black pepper, cover with all but 2 tablespoons of the melted butter and toss to coat.
Pour the remaining 2 tablespoons butter in the bottom of an 8-inch nonstick skillet and swirl it around to coat the bottom and sides. Add the thyme and a pinch of salt. It will start to crackle a little. Cook until it is no longer crackling, about 5 minutes, then remove the thyme from the butter and let cool slightly, about 1 minute. Strip half of the thyme leaves from the stems into the bowl with the potatoes.
Preheat the oven to 350 degrees F.
Assemble: Remember that the bottom layer will be the top when you unmold this cake so it should be done with extra care. Beginning in the center of the skillet, layer the potatoes in an overlapping circle following the edge of the skillet. Continue with the remaining potatoes, switching the direction and adding some salt and add a few spoonfuls of the potato liquid in between each layer. Gently press down on the potatoes as you're layering to make sure they stick together and form a cake. Pour any remaining potato liquid on top.
Cook: Place the skillet on high heat and cook until the water starts to release itself from the potatoes and you can see the edges browning, 10 to 12 minutes. Place the skillet on a baking sheet, transfer the baking sheet to the oven and cook, undisturbed, for 15 minutes.
Remove the baking sheet from the oven and carefully pour off any excess butter from the skillet into a bowl; save. Carefully line the hot baking sheet with parchment paper, then flip the skillet onto the sheet in one deft motion. Lift off the skillet and pour the reserved butter over the cake. Return the baking sheet to the oven to cook until the top of the cake is golden brown and crispy and the potatoes feel completely tender in the center when pierced with the tip of a knife, about 15 minutes.
Serve: Carefully transfer the cake to a cutting board, season with salt and cut into wedges like a pie. Serve with roe or creme fraiche if desired.