Step 1: Activate the Yeast. In a small bowl, combine the warm water and active dry yeast. Allow it to sit for about 5-10 minutes until it becomes frothy.
Step 2: Mix the Wet Ingredients. In a large mixing bowl, combine the lukewarm milk, sugar, salt, eggs, and vegetable shortening. Whisk together until smooth. Once the yeast mixture is frothy, add it to this bowl and mix well.
Step 3: Combine Dry Ingredients. In a separate bowl, whisk together the all-purpose flour, cocoa powder, and red liquid food color. Gradually add this mixture to the wet ingredients, mixing until a dough forms.
Step 4: Knead the Dough. Turn the dough out onto a floured surface and knead for about 5-7 minutes until it is smooth and elastic.
Step 5: First Rise. Place the kneaded dough in a greased bowl, cover it with a clean kitchen towel, and let it rise in a warm place for about 1-1.5 hours, or until it doubles in size.
Step 6: Shape the Donuts. Once the dough has risen, punch it down to release the air. Roll it out to about 1/2 inch thickness. Use a donut cutter to cut out shapes.
Step 7: Second Rise. Cover the shaped donuts with a kitchen towel and let them rise again for about 30 minutes.
Step 8: Heat the Oil. In a deep fryer or heavy pot, heat vegetable oil to 350°F (175°C).
Step 9: Fry the Donuts. Carefully place the donuts in the hot oil, a few at a time, frying for about 2-3 minutes on each side until golden brown.
Step 10: Prepare the Glaze. In a medium bowl, whisk together the confectioners' sugar, milk, and vanilla extract until smooth.
Step 11: Glaze the Donuts. Once the donuts are cool, dip the tops into the glaze, allowing any excess to drip off.