Prepare the yeast mixture. In a small pot, heat the milk until warm but not hot, about skin temperature. Transfer milk to a large bowl and add the yeast and granulated sugar. Stir gently and set aside. Place a stick of butter in the warm pot (off the heat) and allow to melt fully.

Make the dough and allow to rise. Between 5 and 15 minutes, depending on the age of the yeast, you will see tiny bubbles or even clumps -- a sign that your yeast is now ready. Add the melted, cooled butter, eggs, and salt. Stir until incorporated. Add the flour, stirring between each cup to gradually form a sticky, yet firm dough. You may need to grab hold of the bowl firmly as you turn it with a wooden spoon toward the end. Do not knead the dough, simply gather into a ball and cover with a towel or cling wrap. Let rise in a warm place until doubled in size, about 1 hour.

Flatten and cut dough. Turn the dough out onto a lightly floured surface and using your palms flatten into a smooth rectangle about ½-inch thick. Use a dough cutter (a pizza wheel works, too) to cut into one-and-a-half-inch rows by four-inch columns. Meanwhile, slowly bring the oil to temperature in a Dutch oven over medium heat (or medium-high heat depending on the strength of your burner).

Set up a fry station and test the oil. Break off a piece of dough and roll into a ball. Gently drop into the hot oil and watch for bubbles to form around the dough; it will float and turn golden brown when ready. To fry long johns, add up to 3 at a time, using a spider to flip to the other side when golden brown. Strain with the spider and transfer to a paper towel or holding rack until cool enough to handle.

Dip in glaze. In a medium bowl, combine the melted butter, maple syrup, confectioners sugar, maple extract (if using), and a pinch of salt. Stir until smooth. Dip the warm, fried donuts into the icing mixture and set on a serving tray. Allow the icing to form a crystallized smooth layer. Best enjoyed fresh. See notes for storage.