Combine the warm water, granulated sugar, and active dry yeast in a large bowl. Let the mixture stand for 5–10 minutes, until the yeast becomes foamy.

Whisk the beaten eggs, evaporated milk, and fine salt together in a separate bowl. Add this mixture and the shortening to the foamy yeast mixture.

Stir in about 4 cups of flour until combined. Add the remaining flour gradually, stopping when the dough is soft, smooth, and slightly sticky rather than stiff.

Knead the dough by hand for 8–10 minutes, or use a dough hook for approximately 5–7 minutes. Place it in a lightly greased bowl, cover, and let it rise for about 1 hour, until doubled; alternatively, refrigerate it overnight.

Roll the dough on a well-floured surface to approximately 1/4-inch thickness. Cut it into roughly 2 1/2- to 3-inch squares and keep the cut pieces lightly covered while the oil heats.

Heat 2–3 quarts of oil in a deep, heavy pot to 350–360°F (175–182°C). Fry a few squares at a time for about 1–1 1/2 minutes per side, turning once, until puffed and golden brown.

Transfer the hot beignets to a rack or paper towels to drain briefly. Dust them heavily with confectioners’ sugar immediately and serve warm.