In the bowl of a stand mixer fitted with a dough hook attached, add the flour, sugar, salt and yeast and mix until combined.
Add the warm evaporated milk, egg, and 2 tablespoons vegetable oil to the bowl and mix the dough until the dough forms a ball that pulls away from the sides of the bowl.
Place the dough in a lightly greased bowl and cover with plastic wrap.
Let the dough rise in a warm, draft-free place until it doubles in size, 1 to 2 hours.
Heat vegetable oil in a deep fryer until the temperature has reached 350 F. (Alternatively, heat a few inches oil in a Dutch oven.)
While the oil heats, roll out the beignet dough on a lightly floured surface until it is a square about ½-inch thick.
Cut the dough into 2x2-inch squares.
Fry the beignets in the heated oil for 1 to 2 minutes each side, until golden brown.
Remove the cooked beignets and place them on a paper towel lined plate to help absorb excess oil.
Prepare the filling: Add the cream cheese, powdered sugar, vanilla extract and heavy cream to a large bowl or the bowl of a stand mixer and beat until smooth and creamy.
Mix in the crushed cookies until thoroughly combined.
Use a skewer, chopstick, or your finger to make a hole in the side of each beignet and space inside to put the filling.
Use a spoon or a piping bag fitted with a tip to stuff the beignets with the filling.
Sprinkle with additional powdered sugar if desired, then serve.