In the bowl of a stand mixer fitted with the dough hook, combine the water, sugar, yeast, egg + egg yolk, salt, evaporated milk, vanilla, and half the flour (254 grams). Mix until combined. With the mixer on low speed, add the butter and then the remaining flour and increase the speed to medium. Knead until soft, smooth, and slightly tacky but not unbearably sticky, about 2 to 3 minutes. If needed, add more flour 1 tablespoon at a time to bring the dough together.

Place the dough in a greased bowl, and cover with plastic wrap. Let rise in a warm place for about 1 hour, or until doubled in volume. To make ahead: store the risen dough in the fridge for up to 3 days. Let come to room temperature before proceeding with the next step.

Punch down the dough and let it rise again for another 30 minutes.

Preheat 3 to 4 inches of oil in a wide Dutch oven or large heavy-bottomed pot set over medium-high heat to 350°F measured with a deep-fry thermometer. Sift the confectioners’ sugar into a large bowl.

While the oil is heating, roll the dough out on a floured work surface into a rectangle about 1/4-inch thickness, sprinkling with flour if your rolling pin sticks anywhere. Cut into 2 1/2-inch squares with a pastry wheel or pizza cutter.

Deep fry three to four at a time, flipping every 20 seconds, until they become a golden color. Adjust the heat up and down as the oil temperature fluctuates, to keep a consistent temperature of 350°F.

Remove the finished beignets with a slotted spoon or spider, drain them for a few seconds on paper towels, and then toss them into the bowl of confectioners' sugar. Serve immediately.