These are different from the doughnut recipe on page 31 of the study guide.  Those on page 31 are raised doughnuts that require yeast.  These are cake doughnuts that do not have to rise.  The name “sweet milk” means that they use fresh milk, not sour milk like buttermilk.
3 eggs
1 teaspoon vanilla
3/4 cup sugar
3 Tablespoons butter, softened
3/4 cup milk
4 1/2 cups flour
3 1/2 teaspoons baking powder
1 teaspoon salt
1/4 teaspoon nutmeg
1/4 teaspoon cinnamon
Oil for frying
Beat eggs and sugar in a large bowl.  Add vanilla, butter, and milk.  Sift flour, salt, spices and baking powder in an bowl; add to egg mixture and mix well.
Start oil heating in a sauce pan, two inches deep.  Heat the oil over medium heat. 
Roll the dough to 1/3 inch thick.  Cut for doughnuts and check the temperature of the oil.  The oil should 375°F.  (If you don’t have a thermometer, try dropping a single drop of water in the oil.  Stand back, if the oil is hot enough to fry the doughnuts, the water will pop).  Drop the doughnuts into the oil one or two at a time.  Fry until light browned on both sides.  Serve plain or roll in powdered sugar.