In a small saucepan over medium heat, warm the milk until hot but not boiling (about 125°F/52°C on an instant-read thermometer). Remove the pan from the heat.
Fit a stand mixer with the paddle attachment. In the mixing bowl, combine the flour, granulated sugar, salt, and yeast and beat on low speed to mix. Add the hot milk, raise the speed to medium, and beat until blended. Add the butter, whole egg and egg yolk, lemon zest, and vanilla and beat until well blended and smooth, about 2 minutes. The dough will not pull away from the sides of the bowl and will still be somewhat sticky, but your finger should come out clean when you insert it into the center. Scrape the dough into a large bowl, cover, and let rest for about 10 minutes.
Line a baking sheet with waxed paper and brush the paper with oil. Line a second baking sheet with paper towels.
Lightly grease the palms of your hands with oil. Pull off about 1 teaspoon of the dough and roll it between your palms into a smooth ball, then press into a disk about 2 inches (5 cm) in diameter. Place on the oiled paper. Spoon about 1 teaspoonful of the lemon curd into the center of the dough disk. Make a second dough disk the same way and place on top of the first, stretching the edges over the curd. Pinch the edges to seal tightly. Repeat with the remaining dough and lemon curd. You should have about 18 beignets. Cover with a clean kitchen towel and let rise for 30 minutes. The beignets should look soft and puffy, but will not double in size.
Using a spatula, one at a time, transfer 4-5 beignets from the baking sheet to the hot oil, sliding them in gently. Be sure not to overcrowd the pan. The beignets should float to the top and puff to about double their size. Deep-fry until dark golden on the first side, about 1 1/2 minutes. Using tongs, turn and fry until dark golden on the second side, about 1 1/2 minutes longer. Transfer to the towel-lined baking sheet to drain. Repeat to fry the remaining beignets, allowing the oil to return to 365 degrees F between batches.
Arrange the beignets on a platter. Using a fine-mesh sieve, dust with confectioner's sugar and serve right away.
Excerpted with permission from <a href="http://www.amazon.com/gp/product/1740899822/ref=as_li_tf_tl?ie=UTF8&camp=1789&creative=9325&creativeASIN=1740899822&linkCode=as2&tag=craftfocom-20">Donuts by Elinor Klivans</a><img src="http://ir-na.amazon-adsystem.com/e/ir?t=craftfocom-20&l=as2&o=1&a=1740899822" width="1" height="1" border="0" alt="" style="border:none !important; margin:0px !important;" />