Pour the lukewarm water over the dry baker's yeast and stir to rehydrate.
Place the flour, salt and sugar in the bowl of the stand mixer. Stir to combine.
Make a well in the middle of the flour and break in the eggs. Add the vanilla extract and baking powder.
Knead (with hook) on speed 1 to 2 depending on your mixer for 15 minutes. Gradually add the soft butter as soon as you start kneading. You should get a smooth and homogeneous dough that pulls away from the sides of the bowl.
Turn the dough onto a lightly floured surface and knead it quickly to gather it into a ball.
Place it in a bowl and cover with plastic wrap and a towel. Let it rise in a warm place, away from drafts, for 2 hours until it doubles in size.
Once the dough has risen well, turn it onto a floured surface and punch it down by kneading it a little.
Roll out with a rolling pin to a thickness of 4 to 5 mm (square or rectangular to avoid too much leftover dough when cutting)
Using a pastry wheel with a serrated edge, cut strips about 6 cm / 2.4 inches wide, more or less, depending on your preference. Cut the bands into diamonds (cut diagonally every 5 cm / 2 inches or so) Make an incision (slit) in the center of each diamond-shaped piece, which will become the bugnes.
Heat the oil in a frying pan wide enough to avoid splattering.
When the oil starts to shimmer (170°C / 340°F if you have a thermometer), and it is hot, carefully drop the beignets one by one and let them brown.
Once golden on one side, turn them over to brown the other side.
Drain on paper towels and continue frying the remaining bugnes.
Let cool before dusting both sides with powdered sugar.
Enjoy french bugnes lukewarm or cold!