In the bowl of an electric stand mixer, whisk together 2 1/4 tsp active dry yeast, 1/2 cup warm water (110°F) and 1/2 tsp of the 1/4 cup granulated sugar. Let rest 5–10 minutes, until foamy.

Add 1/4 cup evaporated milk (warmed to 110°F), the remaining granulated sugar (the rest of the 1/4 cup), 1/2 tsp salt, 1/4 cup vegetable shortening (room temperature), 1 large egg, 1 egg yolk, and 1/2 tsp vanilla extract. Mix on low just until combined.

Add half of the 2 1/2 cups all-purpose flour to the mixer. With the whisk or paddle attachment, mix until the batter is smooth and the flour is incorporated.

Switch to the dough hook. Slowly add the remaining flour while the mixer runs on low. Knead on low speed about 4–5 minutes, until the dough is smooth and elastic. Add a little more flour if the dough is too sticky to handle.

Transfer the dough to a lightly oiled bowl, turn once to coat, cover with plastic wrap, and let rise in a warm place until doubled in size, about 1 1/2 hours.

Punch the dough down and turn it out onto a lightly floured surface. Roll the dough to slightly less than 1/2-inch thickness.

Cut doughnuts using a doughnut cutter or two round cutters (one large and one small for the holes). Place cut doughnuts and holes on a lightly floured tray or baking sheet, leaving space between them.

Cover the cut doughnuts loosely and let them rise until doubled in size, about 30–40 minutes.

While the doughnuts are proofing, heat 3–4 cups vegetable shortening in a heavy pot or Dutch oven. Use a deep-fry thermometer and bring the oil to 360°F; monitor temperature and do not let it exceed 375°F. (Do not leave hot oil unattended.)

Prepare the glaze in a shallow dish: combine 2 Tbsp unsalted butter (melted), 1 1/3 cups powdered sugar, 1 pinch salt, 2 tsp evaporated milk, and 1/2 tsp vanilla extract. Stir, then add 3–4 tsp hot water a teaspoon at a time until the glaze reaches a fairly thick but dippable consistency. Do not make it runny.

Fry the doughnuts a few at a time (do not overcrowd) in the 360°F oil. Fry until the bottom is golden, then flip with a utensil and fry the other side until golden brown. Adjust heat as needed to keep the oil near 360°F. Transfer fried doughnuts to a wire rack to drain.

Let the doughnuts cool on the rack for 1–2 minutes, then dip the top half of each warm doughnut into the glaze. Return to the rack and allow the glaze to set at room temperature.

Serve doughnuts warm. If they cool and you want them warmed, reheat in the microwave for 5–10 seconds.