Pour the water and milk into a saucepan and gently warm over low heat until it's lukewarm. (It's lukewarm if you can put your finger in it, and it feels pleasant.) Pour into a jug, add the sugar and the yeast, then stir to dissolve. Set aside for about 10-15 minutes to develop and bubble the yeast.
While waiting, melt the butter. Once the yeast is frothing, add the egg, melted butter, cinnamon and vanilla to the milk mixture and whisk to combine.
Combine the flour and salt in a large bowl. Pour in the wet ingredients and stir everything together with a wooden spoon until you have a soft dough.
Tip it out onto a floured surface and knead for 2 minutes until smooth. Shape into a ball and place in a clean greased bowl, cover with cling film or a clean towel and leave to rise in a warm environment for an hour until doubled in size.
Once the dough has risen, tip it out onto a surface, then roll it into a rectangle, around ¼ inch thick. Cut the dough into squares.
Pour the oil into a large pot or saucepan and heat it to 375F/190C. If you don't have a temperature probe, add a piece of bread to the oil. If it sizzles and the bread turns golden in 30 seconds, it's ready. Be careful with the hot oil.
Gently lower a couple of the dough squares into the hot oil using a slotted spoon or spider, and fry them for about 1-2 minutes on each side or until golden brown.
Remove the beignets and let them cool. Repeat with the rest of the beignets.
Using a sieve, cover them in lots of powdered sugar, and then serve!