1/2 cup (1 stick) unsalted butter, melted, plus 2 tablespoons butter for frying
Place potatoes in a bowl and wash under cold, running water until water runs clear. Place in a medium saucepan with 1/2 teaspoon of the salt. Add enough water to generously cover. Bring to a boil, reduce to a simmer, and cook, uncovered, until soft, about 15 minutes. While potatoes are warm, mash with a fork or in a food mill.
Let potatoes cool a little, then mix in the oats to make a soft dough. Add the butter and remaining 1 teaspoon salt and mix well. Roll dough to 1-inch thickness and cut out 3-inch biscuits with a round cutter.
Heat 1 tablespoon butter in a large skillet. Add half of the potato and oatmeal "cakes" and fry until light brown on both sides, about 2 minutes per side. Repeat with remaining cakes, adding more butter to skillet if necessary. Serve warm.