To make clarified butter: Slowly melt the butter in a small heavy saucepan over medium-low heat; do not stir. Remove from the heat and carefully spoon off and discard any foam from the top. Pour the clear liquid butter through a fine-mesh strainer, leaving behind the solids in the pan. Once cool, chill in the refrigerator. The clarified butter can be refrigerated for weeks.
To make the potato dominoes: Preheat the oven to 400 degrees with a rack in the center of the oven. Line a rimmed baking sheet with a silicone mat or use a nonstick baking sheet.
Cut off the 2 ends of 1 potato and reserve them. Trim the 4 sides of the potato to form an even brick. Slice the potato about 1/8 inch thick on a mandoline, keeping the slices in order if you can (just like a line of dominoes). Hold the stack of potato slices in the palm of 1 hand and use the other to shape them back into a brick, as you would a deck of cards.
Lay the stack on its side on the baking sheet, and put the reserved potato ends, cut side down, at either end to keep the stack aligned. Then, with the palm of your hand, angle the slices slightly to resemble a line of dominoes that has tilted over. Adjust the end pieces to keep the stack in shape, and align the slices if necessary. Dot the top and sides with 1 tablespoon of the clarified butter. Sprinkle with salt to taste. Repeat with the remaining potatoes, keeping the stacks at least 2 inches apart.
Bake for 40 minutes, or until the potatoes are browned on the edges and tender in the middle when tested with a skewer. Serve immediately.
Nutrition information per each of 4 servings: