Make the dough: In the bowl of a standing mixer, mix flour, sugar, and yeast. Fit mixer with the dough hook, and add egg, milk, butter, zest, vanilla extract, and brandy to the flour mixture. Mix on low-medium speed until a soft ball of dough starts to form, about 3 minutes. Add in salt. Keep mixing for 8 minutes, until dough is soft and elastic and pulls away from the sides of the bowl. Place dough in a lightly oiled bowl and cover with plastic wrap. Let rest in a warm place for about 2 hours, or until doubled in size.
Gently punch the dough to remove air. On a lightly floured surface, roll dough into 1/2 –inch thickness. Cut into doughnuts using a doughnut cutter or with a 2¾-inch/7 cm round cutter and with a ¾-inch/2 cm round cutter for the holes (or use any cutter that you want or have available depending on your preferred doughnut size). Re-roll scraps and repeat. Place doughnuts on a baking sheet lined with parchment paper. Cover loosely with plastic wrap and let rest to rise for about 60 minutes, or until doubled in size.
Make the cinnamon sugar mixture: Place 8 tablespoons sugar and 1 tablespoon cinnamon in a small bowl and toss to combine. Set aside.
Frying: In a large, deep, heavy-bottomed saucepan, heat 2 in/5 cm of oil until a thermometer inserted into the oil reaches 300F/150C. If you don’t have a thermometer to test, insert a wooden spoon into the oil; it’s ready for frying when tiny bubbles appear around the stick. If it bubbles vigorously, it’s too hot. Carefully add doughnuts, a few at a time, depending on how large the saucepan is. Fry until golden brown, about 2 minutes on each side. Drain doughnuts on a paper towel-lined plate. While doughnuts are still warm, dip them in cinnamon sugar mixture and coat from all sides.
Doughnuts are best the same day they are made.