01 - Dissolve yeast in warm water with a pinch of sugar in a large bowl. Let stand until foamy, about 5 minutes.
02 - Add the remaining sugar, milk, egg, melted butter, and vanilla to the yeast mixture. Whisk until thoroughly combined.
03 - Stir in flour and salt. Mix until a sticky, shaggy dough forms.
04 - Turn the dough onto a floured surface and knead until smooth and elastic, approximately 5 minutes.
05 - Place the dough in a greased bowl, cover with plastic wrap, and let rise in a warm, draft-free place until doubled in size, about 1 hour.
06 - Punch down the dough and turn onto a floured surface. Roll out to 1/2-inch thickness and cut into 2-inch squares. Cover and let rise for 10 minutes.
07 - Heat 2 inches of vegetable oil in a deep pot to 350°F.
08 - Fry the beignets in batches for 1 to 2 minutes per side, until puffed and golden brown. Remove with a slotted spoon and drain on paper towels. Dust generously with powdered sugar while warm.
09 - Heat the cream in a small saucepan over medium heat until just simmering. Remove from heat and add chocolate, butter, vanilla, and salt. Let stand for 2 minutes, then whisk until smooth and glossy.
10 - Serve beignets immediately with warm chocolate sauce for dipping.