Stir the water and yeast together in medium bowl. When well mixed, let sit undisturbed for 10 minutes. After that, stir in 1 cup of the flour until the mixture is smooth. Then, cover the bowl with plastic wrap. Set it aside and allow it to rise in a warm place until bubbly. This will take about 40 minutes.
Beat the butter in a bowl with an electric mixer until creamy. Add sugar and beat until light and fluffy. Add eggs, 1 at a time, beating after each addition. Beat in vanilla and salt. Add yeast mixture and remaining 3 1/2 cups of flour. Then mix until flour is fully incorporated and the dough wraps around the paddle attachment and comes away from the sides of the bowl.
For the glaze, quickly mix together confectioners'sugar, milk, and corn syrup in small bowl. Heat this mixture in the microwave for about 40 seconds to warm.Stir to mix and then quickly brush the glaze over thetops of doughnuts while they are still warm. Allow the doughnuts to cool only long enough for the glaze to firm. Serve immediately. Makes 2 dozen doughnuts.