Heat the milk to about 110°F. Stir in the yeast and let it stand 5 to 10 minutes, until foamy.
In a large bowl, whisk together both sugars, the eggs, egg yolk, melted butter, and maple extract. Pour in the yeast mixture.
Stir in the flour, nutmeg, and salt until a soft sticky dough forms. Knead on a floured surface 2 to 3 minutes.
Place the dough in a greased bowl, cover, and let rise in a warm spot 1 ½ to 2 hours, or until about doubled in size.
Set the dough out on a lightly floured surface and roll it out to ½ inch thick. Cut into 8 rectangles, set on parchment paper and cover while you heat your oil.
Heat ½ inch of oil in a Dutch oven to 350°F. Fry 2 or 3 bars at a time, 1 ½ to 2 minutes per side, until golden. Drain on paper towels and cool.
Whisk the melted butter, powdered sugar, maple syrup, and maple extract until smooth. Dip the top of each bar, then scatter the walnuts over while the glaze is still wet. Let set before serving.
Notes
I make these with my long john donut dough, with just a few changes to turn them into maple bars. Half the sugar is light brown here, maple extract goes in place of the vanilla, and there's a little nutmeg in the dough. If you love these, my long john donuts with fresh milled flour are worth a look too!
I use chopped walnuts on top, but peanuts are the classic maple bar topping, and pecans are lovely here as well, or you can leave the nuts off altogether. They all work perfectly!
Your milk needs to be about 110°F, which feels barely warm when you dip your finger in. If it doesn't foam up within ten minutes, the yeast is too old, and I would start fresh with a new packet.
I cut mine into 8 for a traditional maple bar size, but you can cut them any size you like.
I fry only two or three at a time and keep the oil right around 350°F. Any more in the pot makes the temperature drop, and they come out greasy instead of golden.
These are best the day you make them. Keep them loosely covered at room temperature for 2 days, and unglazed bars freeze for up to 2 months, so thaw them and glaze fresh.