In a small bowl, combine the dry yeast with the warm milk. Stir lightly, then let the mixture sit for 10 minutes to activate the yeast.

In the bowl of a stand mixer fitted with the paddle attachment, beat the softened butter and granulated sugar together until light and fluffy, about 2 minutes.

Add the yeast mixture, egg, egg yolk, salt, and vanilla extract to the bowl. Set the mixer to low speed to combine.

While the mixer is running, gradually add the flour. Continue mixing until the dough becomes smooth. Shape the dough into a round ball, then transfer it to a large, greased bowl. Cover with plastic wrap and let the dough rise in a warm place for about 1 hour, until it has expanded slightly.

Turn the dough out onto a lightly floured surface and roll it into a 1/4-inch thick oval or rectangle. Use a 3-inch biscuit cutter to cut out rounds from the dough. Place the rounds onto a baking sheet lined with parchment paper. Cover loosely with plastic wrap and let them rest for 30 minutes.

Heat the oil in a saucepan until it reaches 350°F, or until a scrap of dough dropped into the oil bubbles immediately. Fry the dough rounds in batches of 2 or 3 at a time, cooking each side until golden brown. Use a slotted spoon to transfer the beignets to a paper towel-lined plate to drain excess oil.

Once the beignets are no longer hot but still warm, dust them generously with powdered sugar and serve immediately.