In the bowl of an electric stand mixer, whisk together yeast, warm water, and 1/2 tsp of the sugar. Let rest for 5 - 10 minutes.

Add in evaporated milk, remaining granulated sugar, salt, 1/4 cup shortening, egg, egg yolk, and vanilla.

Add half of the flour and blend until smooth using the whisk attachment.

Switch to the hook attachment, slowly add remaining flour, and knead on low speed until smooth and elastic, about 4 - 5 minutes, adding additional flour as needed.

Transfer dough to a lightly oiled bowl, cover with plastic wrap, and let rise in a warm place until doubled in size, about 1 1/2 hours.

Punch dough down and roll into an even layer onto a floured surface to slightly less than 1/2-inch thickness.

Cut into doughnut shapes using a doughnut cutter or two round circle cutters (one large and one small for holes). Cover and let rise until doubled, about 30 - 40 minutes.

Heat shortening in a cast iron dutch oven to 360 degrees.

Prepare glaze by mixing together all of the glaze ingredients in a shallow dish.

Carefully transfer doughnuts to oil (fry 3 at a time) and fry until golden on the bottom, then flip and fry the opposite side until golden brown.

Transfer to a wire rack and allow to cool for 1 - 2 minutes, then dip the top half in glaze while still warm and return to wire rack to allow glaze to set at room temperature.