Note: I make this dough the night before and set it in the fridge overnight.
In the bowl of a stand mixer, add the warm water (120g or ½ cup) and warmed milk (½ cup or 120g). Whisk in the active dry yeast (1 ½ teaspoon) and leave for a few minutes so the yeast can dissolve.
Add the brown sugar (67g or ⅓ cup) and whisk to combine. To the bowl add the egg, discard (125g), all purpose flour (500g), vanilla (1 teaspoon) and salt (½ teaspoon). Affix the dough hook and turn the mixer on low as it begins to knead the dough. Once it starts coming together, add the softened butter. Knead until the dough comes around the dough hook in a ball(ish) shape and the bowl is mostly clean of dough.
Transfer to an oiled bowl, toss the dough around to coat with the oil, and leave at room temperature for about 1-2 hours, just enough time for the dough to begin to rise but not enough for it to double in volume. Set in the fridge overnight.
In the morning, take the dough out - if it has doubled it's ready to use. If not, let it sit at room temperature until it’s puffy.
Once puffy, roll it out on a well floured countertop (preferably near the stove so you can transfer the dough easily) until it’s rectangular and a little over a quarter of an inch thick. Use a pizza or pastry roller knife to slice into roughly 2-3 inch squares (depending on how well shaped the rectangle was, some will be bigger and some smaller - this is ok). Cover the dough with a damp cloth and leave to rest for about 30 minutes.
Use a deep frying pan (about 4 inches deep) or a dutch oven to fry the beignets. Set the pan over medium high heat and fill with canola oil until it’s about 3 inches full. Have a thermometer handy, bring the oil to 360 F. Don’t start frying before the oil is this hot or the beignets will absorb too much oil as they struggle to cook.
Lay a few sheets of paper towel over a cookie sheet near you (this is where the beignets will go after frying).
One at a time, carefully drop in a piece of dough - fry only 4 at a time. The top of the dough will tear as it warms and rises. Once the bottom is golden brown, use a spider strainer to flip the beignets so they cook on the other side. If you find you want them darker, you can flip again once the tops (now bottoms) have cooked. Once both sides are a deep golden brown, transfer to the paper towels.
Repeat with the rest of the dough, adding no more than four at a time. Once all the dough is cooked, you can transfer the beignets to a plate and dust with the powdered sugar.
The beignets are best eaten the day of making!