How to Make It
Stir together milk and yeast in the bowl of a stand mixer fitted with the dough hook attachment. Let stand until yeast dissolves, about 1 minutes. Add flour, 1/3 cup sugar, eggs, and salt; beat on low speed until flour is completely incorporated, about 2 minutes. Continue beating until dough is smooth and sticky, 2 to 3 minutes. Add butter, a few pieces at a time, and beat until butter is completely incorporated and dough is soft and elastic (it will still be very sticky), 5 to 6 minutes. Remove dough from bowl; wrap dough tightly in plastic wrap. Refrigerate at least 6 hours or up to 15 hours.
Remove dough from refrigerator, and unwrap; roll dough out on a well-floured surface into a 10-inch square (about 1/2 inch thick). Using a 3 1/2-inch round cutter, cut out 10 donut rounds, re-rolling scrap dough as needed. (Discard remaining dough scraps.) Arrange donut rounds 1 inch apart on a lightly floured baking sheet. Cover baking sheet tightly with plastic wrap. Let donut rounds proof at room temperature until doubled in height, 2 to 3 hours. (Shaped Donuts can also be chilled 12 to 15 hours before proofing.)
Place remaining 1 cup sugar in a small bowl; set aside. Pour oil into a Dutch oven to a depth of 21/2 inches; heat over medium to 375°F. Working in batches, add donuts (be careful not to crowd them); cook until golden brown and a thermometer inserted into donuts registers 200°F, 1 minute and 30 seconds to 2 minutes per side. Transfer donuts to a baking sheet lined with paper towels; let cool about 1 minute. Toss warm donuts, 1 at a time, in bowl of sugar to coat. Set aside on a baking sheet; let cool completely, 30 to 40 minutes.
During final 10 minutes of donut coolinh time, make the maple meringue: Bring maple syrup to a boil in a small saucepan over high. Meanwhile, beat egg whites in a heatproof bowl with an electric mixer fitted with the whisk attachment on medium-high speed until soft peaks form, about 1 minute. With mixer running on medium-high speed, immediately and carefully drizzle boiling syrup down side of the bowl. When all of the syrup has been added, increase mixer speed to high. Add salt; beat until mixture is light, fluffy, and cooled to room temperature, 3 to 4 minutes. Use immediately.
Using a 1/3-inch round piping tip, poke a 1/2-inch-deep hole into one side of each donut. Transfer maple meringue to a piping bag fitted with 1/3-inch round piping tip. Working with 1 donut at a time, insert piping tip into the premade donut hole; fill with about 1/3 cup maple meringue. Serve filled donuts immediately.