Dissolve yeast and 1 teaspoon white sugar in 1/2 cup warm water in a small bowl. Let stand until it forms a creamy foam layer on top, 5 to 10 minutes.

Heat milk and vegetable shortening in a small saucepan over medium-low heat until shortening melted and tiny bubbles appear around edges. Remove from heat.

Place 1/4 cup white sugar and salt into the bowl of a stand mixer fitted with the paddle attachment; add milk mixture. Beat on low speed until sugar is just dissolved; let stand until warm.

Beat in 1 cup flour, yeast mixture, and eggs on low speed until smooth. Gradually beat in remaining 3 cups flour until forms a soft dough.

Continue beating dough on low speed until it pulls away from the sides of the bowl, 2 to 3 minutes.

Turn out dough onto a well-floured work surface; knead until smooth and elastic, 2 to 3 minutes.

Form dough into a ball; place into a large greased bowl and grease top of dough. Cover the bowl with waxed paper and let dough rise a warm place until doubled in volume, 1 to 1 1/2 hours.

Punch down dough; roll out on a floured work surface until 1/2-inch thick. Cut out doughnuts with a 3 1/2-inch round doughnut cutter with a 1-inch center cutter. Let doughnuts rise on the work surface until very light and fluffy, 40 to 60 minutes.

Heat oil in a deep fryer or large saucepan to 350 degrees F (175 degrees C).

Gently pick up and carefully drop doughnuts into the hot oil in batches of 2 or 3; fry until light golden brown, flipping once, about 2 minutes. Repeat with remaining doughnuts.

Transfer doughnuts to paper towels to drain.

Combine 1 tablespoon warm water and vanilla extract in a bowl; stir in confectioners' sugar to make the glaze. Spread a thin layer of glaze over warm doughnuts.