Make the yeast mixture: Combine yeast, 1/2 cup warm water, and 1 tsp. granulated sugar in bowl of a heavy-duty stand mixer; let stand 5 minutes. Add milk, eggs, salt, and remaining granulated sugar.
Form a dough: Microwave remaining 1 cup water until hot (about 115°); stir in shortening until melted. Add to yeast mixture. Beat at low speed, gradually adding 4 cups flour, until smooth. Gradually add remaining 2 1/2 to 3 cups flour, beating until a sticky dough forms. Transfer to a lightly greased bowl; turn to grease top. Cover and chill 4 to 24 hours.
Roll and cut: Turn dough out onto a floured surface; roll to 1/4-inch thickness. Cut into 2 1/2-inch squares.
Fry until golden: Pour oil to depth of 2 to 3 inches into a Dutch oven; heat to 360°. Fry dough, in batches, 2 to 3 minutes on each side or until golden brown. Drain on a wire rack. Dust immediately with powdered sugar.