In a large, heavy-bottomed pot, heat 2-3 inches of oil to 370 degrees. (I aim for 380 because, by the time you add your donuts in, the temperature will drop to the right where you want it at 370.)
In a large bowl, whisk** flour, salt, leavenings, spices, and brown sugar together to combine. Create a well in the center of the dry ingredients. Pour in remaining wet ingredients to the well and whisk together to break apart the eggs. Then start incorporating all the dry ingredients until a thick dough forms. It might take 2-3 minutes to incorporate all the dry ingredients. Refrigerate dough 10-20 minutes.
Flour a board (or clean counter) and dough well and roll out dough to 1/2-inch thickness. Use a donut cutter to create 3-inch donuts and fry 2-3 at a time: brush excess flour off of cut donuts and slide into hot oil gently. Fry 30 seconds per side (15-20 seconds for donut holes), and drain on paper towels until cool enough to handle. Watch your thermometer as you fry and adjust as you go.
For the glaze, whisk powdered sugar together with vanilla, melted butter, and enough milk to create a glaze slightly thinner than white school glue. Dip warm donuts into glaze and place on cooling rack to drip and dry. The glaze won't stay white on the donut if you dip them warm, so if you'd like a more obvious glaze, either re-dip them again into the glaze once completely cooled, or wait until they're completely cooled to begin with.
Store in airtight containers up to 24 hours. After that, they *just* aren't the same. Still good, but not glorious.
*If you don't have or know what half & half is, substitute 2 tablespoons milk and 2 tablespoons heavy cream to create your own half & half. In a pinch, you can use 2 tablespoons heavy cream and 2 tablespoons cold water OR 3 tablespoons milk.
**For best results, stir everything by hand!