Start by pouring 1 cup of warm water into a mixing bowl. Sprinkle the 2 teaspoons of active yeast and 1 tablespoon of sugar over the water. Allow it to sit for about 5-10 minutes until it becomes frothy.
In another bowl, whisk together the 2/3 cup of whole milk, 1 large egg, 1 egg yolk, and 3 tablespoons of melted unsalted butter. Once combined, add the frothy yeast mixture to this bowl and mix until well combined.
In a separate large bowl, whisk together 4 1/3 cups of all-purpose flour and 1 teaspoon of sea salt. Create a well in the center and pour in the wet ingredients.
Using a wooden spoon or your hands, mix until the dough begins to come together. It should be slightly sticky but manageable.
Transfer the dough to a floured surface and knead it for about 5-7 minutes until smooth and elastic.
Place the kneaded dough in a lightly greased bowl, cover it with a clean kitchen towel, and let it rise in a warm place until it doubles in size, approximately 1-2 hours.
Once the dough has risen, punch it down to release air bubbles. Roll it out on a floured surface to about 1/4 inch thick.
Using a sharp knife or pizza cutter, cut the dough into squares, roughly 2-3 inches in size.
In a deep fryer or large pot, heat enough neutral oil to submerge the beignets (about 2-3 inches deep) to 350°F (175°C).
Carefully drop the squares of dough into the hot oil, a few at a time. Fry them until they puff up and turn golden brown, about 2-3 minutes per side.
While the beignets are still warm, generously dust them with powdered sugar.
