Bloom the yeast: In a large bowl, stir the yeast into the warm water with 1 tsp of the measured granulated sugar. Let stand until foamy, 5–10 minutes.
Make the wet base: In a separate bowl, whisk the evaporated milk, egg, remaining granulated sugar, kosher salt, and melted butter until smooth.
Combine: Pour the milk mixture into the yeast mixture and whisk to combine.
Add flour and knead: Add about half the flour and stir to form a thick batter. Add the remaining flour and mix until a shaggy dough forms. Turn out and knead until smooth and slightly tacky (not dry), 3–5 minutes, resisting the urge to add extra flour.
First rise: Place the dough in a lightly oiled bowl, cover, and let rise at warm room temperature until doubled in size, 1–1.5 hours.
Chill to firm: Gently deflate, re-cover, and refrigerate until slightly firm and easier to roll, at least 1 hour.
Heat the oil: Pour the vegetable oil into a heavy 5–6 qt pot to a depth of about 2 inches and heat to 360–370°F, maintaining 365°F as you fry.
Roll and cut: On a lightly floured surface, roll the chilled dough to 0.25 inch thick. Cut into 2–3 inch squares or diamonds, gathering and rerolling scraps once.
Fry: Fry 3–4 pieces at a time, turning once or twice, until puffed and deep golden brown, 1–2 minutes per side. Adjust heat to keep the oil near 365°F.
Drain and sugar: Transfer beignets to a rack or paper towels to drain briefly, then dust generously with confectioners' sugar while hot.
Serve: Pile onto a warm platter and serve immediately, with extra confectioners' sugar at the table if desired.