Bring 1 1/4cups water and the butter to a boil in a saucepan. Remove from the heat. Mix the flour and the cornstarch together and stir into the liquid until to a smooth ball forms. Return the pan to the stove and stir over low heat until a white film appears on the bottom of the pan.
Transfer the mixture to a mixing bowl. Combine with the sugar, vanilla, eggs, and baking powder, using an electric mixer.
Cut parchment paper into 3-inch squares. Spread them out on a clean work surface and grease them well. Put the dough into a pastry bag with a large serrated spout and squeeze a circle of about 2 inches on each piece of parchment paper.
Heat the oil to 370 degrees F in a deep-fryer or in a large saucepan. The oil needs to be hot enough to sizzle a bread crumb. Let the crullers glide into the oil by carefully turning the parchment squares upside down and gently rolling up the two opposite edges. Fry only a few crullers at a time. They should be golden brown and puffy. Remove from the oil and drain on paper towels.