Crispy and Creamy Doughnuts

Name: Crispy and Creamy Doughnuts
Prep time: 10 min
Cook time: 30 min
Total time: 40 min
Cuisine: American
Ingredients: 2 (.25 ounce) envelopes active dry yeast; 0.25 cup warm water (105 to 115 degrees); 1.5 cups lukewarm milk; 0.5 cup white sugar; 1 teaspoon salt; 2 eggs; 0.33333334326744 cup shortening; 5 cups all-purpose flour; 1 quart vegetable oil for frying; 0.33333334326744 cup butter; 2 cups confectioners' sugar; 1.5 teaspoons vanilla; 4 tablespoons hot water or as needed