Bring potatoes to a boil in a pot of lightly salted water. Reduce heat, and simmer until fork tender, 10 to 12 minutes. Drain well.
While still warm, push potatoes through the large holes of a ricer, or use a potato masher to mash them until smooth (you should have 2 1/2 cups). Stir butter into warm potatoes until combined well. Stir together flour, baking powder, 1 teaspoon salt, and 1/4 teaspoon pepper, then stir into potatoes with a wooden spoon until just combined. Form dough into a ball on a lightly floured work surface. Cover with a dish towel, and let cool for 20 minutes.
Dust rolling pin and work surface with flour, and roll out dough to an 8-by-10-inch rectangle. Sprinkle half with cheese, and fold to create a 4-by-10-inch rectangle. Gently roll out. With a floured knife, cut dough into four 2 1/2-by-4-inch rectangles, then cut each in half diagonally.
Heat a griddle or a large cast-iron or non-stick skillet over medium heat, and add 1 teaspoon butter. When it has completely melted and is sizzling, cook 4 scones until golden brown and cheese melts, 3 to 4 minutes per side. Keep scones warm in a 200-degree oven while you cook the remaining dough.