Mix dry ingredients together in the bowl of a stand mixer fitted with the hook attachment. Add the liquid ingredients and mix until well combined. Cover with a towel and let it rest for 8 minutes.
Knead the dough on medium speed for about 5 minutes until a soft smooth dough forms. Place dough in a greased bowl and then turn it so the top of the dough is wet with oil. Cover the bowl with a lid or towel and let rise for 1 1/2 to 2 hours or until double in bulk.
Punch the dough down and empty onto a floured surface and roll out to about 1/3 inch thick and use a biscuit cutter 2 1/2 to 3 inches wide to cut out donuts. Knead scraps together and roll out again and cut until no longer enough dough. Place on parchment lined baking sheets, cover lightly with plastic wrap, and let rise for 30-60 minutes until almost double in size.
Pour oil in a pot until 2 inches deep and heat over medium until 350 degrees F. Fry dough, about 3 at a time until golden and then flip. Should take 30-60 seconds per side. Drain on a paper towel lined baking sheet.
Once cooled enough to handle, dip the most rounded side of the donut in the frosting and then set on cooling rack.
For the Frosting:
In a small saucepan, heat butter, brown sugar, half & half, and corn syrup over medium heat until sugar is dissolved, Turn to low and whisk in maple extract, salt, and 1/2 cup powdered sugar at a time until smooth. Remove from heat.
The frosting does start to dry and harden which is what you want but just give it a quick stir before you dip each donut. If it starts to get too thick, set the pan back over low heat and whisk in a teaspoon more of half & half.