Stir the yeast into the warm water and let sit for 5 minutes.
In a mixing bowl, combine milk, sugar, and salt. Using a mixer with a paddle attachment, beat in eggs one at a time, stopping to scrape down the sides of the bowl after each addition.
Beat in shortening and 2 cups of flour.
In a small bowl, stir together cocoa powder and food coloring to make a paste.
Add food coloring paste to dough and mix well.
Keep adding flour, 1/2 cup at a time, until dough begins to pull away from sides of bowl.
Switch to a dough hook and knead dough for 5 to 8 minutes, until soft but not sticky.
Place dough in a lightly oiled bowl, turning dough to coat it in oil.
Cover bowl with plastic wrap and let rise until it doubles in bulk, about 1 1/2 hours.
Fill a Dutch oven with about 2 inches of oil and heat to 350 degrees.
Cut dough into four pieces, and working 1 piece at a time, roll dough out on a lightly floured surface to a thickness of 1/4-inch. Use a 3-inch round cookie cutter and a small round cutter to cut donuts. You can use the small round cutter to make more donut holes out of the scraps.
Carefully slide a few donuts into the oil. Fry for 35 to 45 seconds and then flip over. Cook the second side about 30 seconds and then place on a rack or a paper towel-lined baking sheet.
For glaze, combine confectioners' sugar, milk, and vanilla extract in a bowl and whisk until smooth.
Drizzle glaze on top of dounuts.