Bloom the yeast: In the bowl of a mixer or a large bowl, whisk warm milk and granulated sugar, then sprinkle the yeast over the top. Let sit until foamy, about 5-10 minutes.
Make the dough: Whisk in eggs, softened butter, and vanilla. Add flour, salt, and nutmeg. Mix with a dough hook or by hand until a soft dough forms, then knead until smooth and elastic, about 5-8 minutes. The dough should be slightly tacky but pull away from the bowl; add a spoon of flour if overly sticky.
First rise: Place the dough in a lightly greased bowl, cover, and let rise in a warm spot until doubled in size, about 60-90 minutes.
Shape bars: Turn the dough onto a lightly floured surface and roll to about 0.5 inch thick. Cut into 12 rectangles about 1.5 by 5 inches. Transfer to parchment-lined trays, cover lightly, and proof until puffy, about 30-45 minutes.
Heat the oil: In a heavy pot or fryer, heat 2-3 inches of oil to 350 F. Maintain temperature between 340-360 F.
Fry: Carefully lift each bar and fry 1-2 minutes per side until deep golden and cooked through. Transfer to a wire rack set over a sheet pan to drain.
Make the maple glaze: In a saucepan over low heat, melt butter. Whisk in maple syrup and milk; warm until steaming, not boiling. Remove from heat, then whisk in powdered sugar, maple extract, and salt until smooth. Adjust with a splash of milk for a pourable glaze.
Glaze the bars: While the donuts are still warm, dip the tops into the maple glaze or spoon glaze over them. Let excess drip off and set for a few minutes before serving.
Notes
For the lightest texture, avoid adding too much flour; the dough should be soft. Keep the oil temperature steady to prevent greasy donuts. Best enjoyed the day they are made; store leftovers covered at room temperature up to 1 day.