Step 1

Heat milk in a small saucepan over medium just until warm to the touch (110°–115°). Transfer to a heatproof measuring glass. Top off with water to bring volume back to ¾ cup if needed. Immediately stir in yeast and a pinch of sugar; let sit until mixture begins to bubble and foam, about 4 minutes.

Step 2

Mix flour, salt, and remaining ¼ cup sugar in the bowl of a stand mixer fitted with the dough hook on low speed to combine. With motor running, pour in milk mixture, then add eggs one at a time, followed by egg yolks one at a time, mixing well after each addition. Continue to mix, scraping down sides of bowl as needed, 3 minutes. (Dough should be workable but somewhat sticky at this point. You can add a little more flour by the tablespoon if needed, but resist the urge to add too much; you don’t want your dough to be tough.) Add 6 Tbsp. butter, a piece or two at a time, mixing after each addition until incorporated. (You might have to stop the mixer and knead butter in by hand to get it started.) Increase speed to medium and mix until dough is smooth, supple, and elastic (it will still be a little sticky), 6–8 minutes. (Alternatively, if you don’t have a stand mixer, mix dough with a wood spoon in a large bowl, then knead in bowl about 6 minutes, dusting with additional flour if needed. Turn dough out onto a lightly floured surface and continue to knead by hand until smooth.)

Step 3

Lightly butter a large bowl and transfer dough to bowl. Cover with plastic wrap or a kitchen towel and let sit in a warm, draft-free spot until doubled in size, about 1 hour. (Alternatively, you can cover and chill at least 3 hours and up to 12 hours.)

Step 4

Turn out dough onto a lightly floured surface and roll out to ½" thick. Line 2 rimmed baking sheets with kitchen towels; dust towels well with flour. Punch out as many dough rounds as you can with 4"-diameter cutter, then cut out centers with 1"-diameter cutter. Transfer doughnuts and holes to prepared baking sheets. Gather scraps, reroll, and punch out additional rounds; transfer to baking sheets (you should have 10 doughnuts and 10 holes). Lightly cover with plastic wrap and let sit in a warm, draft-free spot until nearly doubled in size and dough springs back slowly when gently pressed, at least 30 minutes and up to 2 hours, depending on temperature of your room (watch the dough, not the clock).

Step 5

Just before you are ready to fry doughnuts, cook maple syrup in a small skillet over low heat until reduced by half, 6–10 minutes.

Step 6

Whisk powdered sugar, salt, and 1 Tbsp. milk in a small bowl until smooth. Whisk in reduced maple syrup and another 1 Tbsp. milk if needed. Glaze should be about the consistency of heavy cream.

Step 7

When ready to fry, line 2 rimmed baking sheets (or a few plates) with paper towels. Fit thermometer to sides of a medium heavy pot and pour in oil to come 2" up sides. Heat over medium until thermometer registers 350°. Working with 2 doughnuts at a time and adjusting heat as needed to maintain temperature of oil, carefully slide doughnuts into oil with a spider or slotted spoon and cook until golden brown, about 2 minutes per side. Transfer to prepared baking sheets and let cool slightly. Working in 2 batches, repeat process with doughnut holes.

Step 8

Dip tops of warm doughnuts and holes in maple glaze and transfer to wire racks. Let sit to allow the glaze to set slightly before serving, about 3 minutes.