In a large measuring cup or bowl, combine warm water, warmed milk, 2 teaspoons of yeast, and one tablespoon of sugar. Stir to combine and sit for 10 minutes until the mixture is bubbly and foamy.
In a separate bowl, whisk together flour, cocoa powder, and salt. Set aside.
In the bowl of a stand mixer, whisk the egg and egg yolk until combined. Add the vanilla extract and whisk again.
Add half of the dry flour mixture to the egg mixture. Using the dough hook attachment, mix on low until just combined.
Pour in the activated yeast mixture and continue mixing with the dough hook until a wet dough forms. (It will still be sticky—this is normal at this stage.)
Add the rest of the flour mixture and knead for about 3 minutes, until a thick dough forms.
Pour in the melted butter and knead again until the dough becomes smooth, soft, and evenly mixed.
Transfer the dough to a lightly greased or clean bowl. Cover with plastic wrap and refrigerate for 2 hours. The dough should double in size.
Once risen, transfer the dough to a floured surface. Roll out to ½-inch thickness and cut into 2x2-inch squares using a knife or pizza cutter.
In a deep pot, heat oil to 350°F. Line a plate with paper towels to drain fried beignets.
Fry beignet squares in batches, about 1 to 1 ½ minutes per side, until puffy and dark brown. Don’t overcrowd the pot. Transfer cooked beignets to the paper towel-lined plate.
In a bowl, whisk together powdered sugar and cocoa powder. This will be your dusting sugar.
Optional: Drizzle chocolate sauce over the warm beignets.
Then sift the cocoa powder and sugar generously over the top. The sweetness comes from this final coating, so don’t be shy!
Serve warm and enjoy every fluffy, chocolatey bite!