In a small bowl, combine the warm water and active dry yeast. Allow it to sit for about 5-10 minutes until it becomes frothy.
In a large mixing bowl, whisk together the frothy yeast mixture, granulated sugar, eggs, whole milk, and vanilla extract until well combined.
In another bowl, whisk together the all-purpose flour and melted salted butter. Gradually add this mixture to the wet ingredients. Stir until a soft dough begins to form.
Transfer the dough to a lightly floured surface. Knead the dough for about 5-7 minutes, adding more flour if necessary, until it is smooth and elastic.
Place the kneaded dough in a greased bowl, cover it with a clean kitchen towel, and let it rise in a warm place for about 1-2 hours or until it has doubled in size.
Once the dough has risen, punch it down to release the air. Turn it out onto a floured surface and roll it out to about 1/4-inch thick. Cut the dough into squares or rectangles, about 2-3 inches in size.
Place the cut dough pieces on a floured baking sheet, cover them with a towel, and let them rise for another 30 minutes.
In a deep pot or Dutch oven, heat the vegetable oil to 350 degrees F.
Carefully drop a few dough pieces into the hot oil, making sure not to overcrowd the pot. Fry them for about 2-3 minutes on each side, or until they are golden brown. Use a slotted spoon to remove them and transfer them to a cooling rack lined with paper towels.
While the beignets are still warm, generously dust them with powdered sugar. Serve immediately and enjoy!