Mix using the straight dough method: Place the eggs, vanilla paste, and milk in the bowl of an electric mixer first, then the flour, yeast, sugar, and salt, and finally the butter on top. Mix on medium speed using a dough hook until full gluten development is achieved.
Let the dough bulk ferment at room temperature for 30 minutes.
Fold the dough onto itself and ferment for another 30 minutes.
Roll the dough to .5 cm / .25 in thick. Relax the dough for 30 minutes.
Score half of the dough using a 1.25-cm / .5-in round cutter. Egg wash the scored half with the egg yolks.
Place a small amount (about 1 g) of chocolate where the dough was scored.
Fold the clean half of the dough onto the chocolate half. Press down with your hands so you can see where the chocolate sits.
Using the dull side of the same 1.25-cm / .5-in cutter, score the dough where the chocolate sits. This is just to make sure that the dough has adhered. Using a 2.5-cm / 1-in cutter, cut around the chocolate.
Proof at 29°C / 84°F for 20 to 25 minutes, or until the dough has doubled in size.
While the beignets proof, heat the oil in a fryer to 180°C / 360°F.
Fry the beignets in the canola or peanut oil until they are golden brown, about 2 minutes, stirring constantly as they fry to ensure proper coloring.
As soon as they come out of the fryer, toss them in the sugar. Serve immediately.