Sift together the flour, baking powder, baking soda, salt and cinnamon in a large bowl. In the bowl of an electric mixer fitted with a paddle, beat the eggs until foamy, gradually add the sugar and continue to beat until thick and yellow. Melt butter in a saucepan over low heat and combine with milk and vanilla extract. Stir the milk mixture into the egg mixture until blended. Add the dry ingredients to the wet ingredients and mix just until incorporated. Chill for 30 minutes to make it easier to roll and cut.
Roll or pat the dough out on a heavily floured surface to about 1/4-inch thick, the dough is somewhat wet. Cut with a floured doughnut cutter, saving the holes. Transfer to a sheet of waxed paper and allow to air dry for 10 minutes. The dough will form a slight crust and absorb less fat when fried.
Heat 3-inches of vegetable oil or shortening to 375 degrees F in an electric fryer or deep saucepan. Fry doughnuts until golden, about 5 minutes each side. To keep the oil temperature constant, fry 3 at a time. Fry the holes separately and drain on paper towels.
For the chocolate glaze: melt butter and chocolate in a saucepan until melted, remove from heat. Whisk in confectioners' sugar until smooth. Gradually stir in the water until the mixture reaches the desired consistency. Dip the tops of the doughnuts in the glaze.
*Variation: for chocolate donuts, melt 3 ounces of unsweetened chocolate along with the butter and continue as directed
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