Traditional Irish potato pancakes, also known as boxty, are made with a mixture of mashed and grated potatoes for a texture that's part pancake, part hash brown. Serve with Irish Bangers and Sauteed Swiss Chard for a complete St. Patrick's Day meal.
Make the mashed potatoes: Bring a saucepan of water fitted with a steamer insert to a boil over medium-high heat. Peel and cut potatoes into 1-inch pieces; add to steamer insert. Cover and cook until fork-tender.
Heat milk and butter in a small saucepan over medium-high heat. Transfer potatoes to a large bowl; mash potatoes using a potato masher. Slowly add milk mixture and stir to incorporate with a wooden spoon until smooth; season with salt and pepper.
Make the boxty potatoes: Transfer 1 1/2 cups mashed potatoes to a large bowl; reserve extra mashed potatoes for another use. Add grated potato to bowl with mashed potatoes. Whisk together egg and egg yolk; add to bowl with potatoes along with milk and scallions and stir to combine. In a medium bowl, whisk together flour, salt, baking powder, and pepper; add to bowl with potatoes and stir to combine.
Melt 2 tablespoons butter in a large heavy-bottomed skillet over medium heat. Working in batches, adding butter as necessary, add 2 tablespoons batter to skillet and gently spread to form a 2-to-3-inch round; cook, turning, once, until browned on both sides, about 4 minutes per side. Serve immediately with butter.