Place the flour, sugar, baking powder, and salt in a food processor and mix for 10 seconds.
Add the butter and mix 10 seconds more.
Add the ice water and mix 30 seconds more or until the mixture leaves the sides of the bowl and forms a ball.
Wrap in plastic and refrigerate 30 to 45 minutes to rest. This resting time will let you roll the dough very thin with little resistance.
Cut the cheese into sticks about 1/3 by 1/3 by 4 inches.
Take the ball out of the refrigerator. Sprinkle some flour over the work surface. With a rolling pin, roll the dough out to a rectangle about 16 by 12 inches. Cut into 1-inch wide strips, and brush the dough with egg.
Wrap each strip around each piece of cheese, and carefully press on the ends so that they are well sealed. When you finish, roll each stick between your hands to be sure they are well sealed. At this point, you can place them in a baking sheet and set them in the freezer. Once they are complelety frozen, save them in plastic bags and fry them frozen.
In a heavy deep pan, heat the oil to 350º F degrees. Carefully drop the deditos into the oil one by one, so that they don’t stick together. Fry for 30 to 40 seconds on each side or until golden.
Remove from the oil with a slotted spoon and place on paper towels to drain. Serve immediately.