The thing about trying to cook meals on a budget, they often end up making me feel like I didn't make anything special. Then, I put a call in to my inner chef and realized that with a few ingredients and a little elbow grease, I can make something that feels fancy and exciting without breaking the bank.
In a small saucepan, melt the butter 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.
Using a mandolin slicer or a sharp knife, cut all of the potatoes into thin (1/8-inch thick) slices. Transfer them to a bowl and cover them with 3/4 of the melted butter. Season with a little salt and the dried thyme and toss to coat the potatoes with the butter. Pour the remaining butter in the bottom of the cast iron skillet and swirl it around to coat the bottom and sides.
Preheat the oven to 400 degrees F.
Remember that the bottom layer with be the top when you unmold this cake so it should be done with extra care. Layer the potatoes, following the edge of the skillet, so they overlap. Start the second circle, closer to the center, of overlapping potato rounds. Continue to make circles with the potatoes until the entire bottom of the skillet is filled with potato rounds in smaller and smaller circles. Fill the skillet with another level layer of potatoes. Sprinkle a touch of salt and make 3 more layers. Press down gently on the potatoes to make sure they are starting to stick together and form a cake.
Place the skillet on high heat and cook until the water starts to emerge from the potatoes and you can see the edges browning, 5 to 8 minutes. Place the skillet in the oven and cook, undisturbed, until the potatoes feel tender in the center when pierced with the tip of a knife, 10 to 15 minutes. Remove the skillet and carefully pour off any excess butter or liquid into a bowl. Place the skillet on a flat surface and cover it with a platter larger than the skillet. Carefully turn the platter over in one deft motion. Lift off the skillet and use a large metal spatula to place it back in the skillet so it can brown on the second side. Pour the butter back in and cook in the oven for an additional 5 to 8 minutes. Remove from the oven. Pour off any liquid. Season with salt. Cut into wedges like a pie. Serve immediately in the skillet. Alternatively, lift it out of the skillet and serve on a platter or in slices on individual plates.