Instructions

Place 1½ pounds russet potatoes, peeled and cut into 1-inch pieces, in medium saucepan and add water to just cover. Bring to boil over high heat; reduce heat to medium-low and simmer until potatoes are tender when pierced with fork, about 10 minutes.
Drain potatoes and return to saucepan. Off heat, use potato masher to mash potatoes until no large lumps remain. Measure 2 very firmly packed cups (1 pound/454 grams) potatoes and transfer to large bowl. Stir in 4 tablespoons unsalted butter, cut into ½-inch pieces, until melted, mashing any remaining potato pieces. Whisk in 1 cup canned sweetened condensed milk and 1 cup (7 ounces) granulated sugar. Whisk in 3 large eggs until combined.
Whisk 3¾ cups (18¾ ounces) all-purpose flour, 1½ tablespoons baking powder, and 1 teaspoon table salt in second bowl. Using silicone spatula or wooden spoon, stir flour mixture into warm potato mixture until all flour is moistened and dough resembles thick cake batter. Refrigerate, uncovered, until dough is chilled, 1 hour or up to 3 hours.
Line rimmed baking sheet with parchment paper and dust with flour. Dust counter heavily with flour. Turn out dough onto floured counter, then dust top of dough with additional flour. Using your floured hands, gently pat dough into ½-inch-thick round, about 13 inches in diameter.
Using floured 3-inch round cutter, cut 1 doughnut. Using floured 1¼-inch round cutter, cut hole in center of doughnut. Transfer doughnut and hole to prepared sheet. Repeat cutting remaining dough, 1 doughnut at a time, reflouring cutters between cuts (you should have 12 of each; doughnuts can be touching on sheet). Gather dough scraps, form into ball, dust counter and top of dough with flour, and pat ½ inch thick. Repeat cutting to make 3 more doughnuts and holes. Refrigerate, uncovered, for 1 hour or up to 3 hours.
About 20 minutes before frying, add 2 quarts vegetable oil to large Dutch oven and heat over medium heat to 350 degrees. Set wire rack in second rimmed baking sheet and line with triple layer of paper towels. Using both hands, gently place 5 doughnut rounds in oil. Cook until deep golden brown on undersides, 1½ to 2 minutes, adjusting burner as necessary to maintain oil temperature between 325 and 340 degrees. Using spider skimmer or slotted spoon, flip doughnuts and cook until second sides are deep golden brown, about 1½ minutes.
Using spider skimmer or slotted spoon, transfer doughnuts to prepared rack. Return oil to 350 degrees and repeat with remaining doughnut rounds.
Return oil to 350 degrees and, using spider skimmer or slotted spoon, carefully add doughnut holes to hot oil. Cook, stirring often, until deep golden brown on all sides, about 2 minutes. Transfer to prepared rack. Let doughnuts cool, about 15 minutes.
While doughnuts cool, whisk 4 cups (16 ounces) confectioners’ sugar, 6 tablespoons hot water, remaining 2 tablespoons sweetened condensed milk, and remaining ¼ teaspoon salt in medium bowl until smooth. Adjust consistency with extra hot water as needed, 1 teaspoon at a time, until glaze has consistency of runny craft glue. Working with 1 doughnut at a time, dip both sides in glaze, allowing excess to drip back into bowl. Return to rack and repeat with doughnut holes. Let doughnuts and holes stand until glaze has become slightly matte and dry to touch, 15 to 30 minutes, before serving. (Doughnuts can be stored in airtight container at room temperature, stacked between layers of parchment paper, for up to 2 days.)