Set the oven rack to the middle position and preheat the oven to 200°F (95°C). Line a baking sheet with a wire rack and set aside.

In a medium bowl, whisk together the flour, 1 teaspoon salt, baking soda, and black pepper. Set aside.

Peel 1 pound of the potatoes and cut them into 1/2-inch pieces. Place them in a medium saucepan with 1 teaspoon salt and enough cold water to cover by 1 inch. Bring to a boil over high heat, then reduce to medium and simmer until very tender when pierced with a fork, about 15 minutes. Drain and let cool in a colander for about 10 minutes. Transfer to a large bowl and mash until smooth with no lumps.

While the boiled potatoes cool, peel the remaining 1 pound of potatoes. Using a box grater placed over a clean kitchen towel, grate the potatoes onto the towel. Gather the towel and squeeze out as much liquid as possible over the sink. Add the shredded potatoes to the bowl with the mashed potatoes and mix gently. Stir in scallions and egg until combined. Add the flour mixture and mix well. Using a spatula, add the buttermilk and gently stir until the batter becomes thick and pancake-like with no visible dry flour.

Heat 1 1/2 tablespoons butter in a large nonstick or cast iron skillet over medium heat until melted and bubbling. Scoop four heaping 1/4-cup portions of batter into the skillet and flatten into circles about 4 inches wide and 1/2 inch thick. Cook without moving until the bottom turns golden brown, about 4 to 6 minutes. Carefully flip and cook the other side for another 4 to 6 minutes until golden. Transfer the pancakes to the prepared wire rack on the baking sheet and place in the oven to keep warm. Repeat with the remaining batter in two more batches, adding 1 1/2 tablespoons butter between batches.

Before serving, sprinkle the hot potato pancakes with the remaining 1/4 teaspoon salt and garnish with additional scallions. Serve with optional toppings such as sour cream, shredded cheddar cheese, cooked bacon, or fried eggs.