Preheat oven to 350 degrees F (175 degrees C). Grease a 13x9-inch baking dish.
Unroll one of the cans of crescent dough and place into prepared baking dish; press onto bottom of pan to form a crust, firmly pressing seams together to seal.
Beat cream cheese, 3/4 cup confectioners' sugar, egg white, and vanilla extract with an electric hand mixer on medium speed in a bowl until well blended; spread onto crust. Spread cherry pie filling over the cream cheese mixture.
Pat remaining crescent dough onto a large sheet of waxed paper forming a 13x9-inch rectangle, pressing seams together to seal. Invert over pie filling to form top crust; discard waxed paper.
Bake in the preheated oven until golden brown, 30 to 35 minutes; cool at least 20 minutes.
Stream milk into remaining 3/4 cup confectioners' sugar in a bowl, while beating until well blended and thick; drizzle over warm dessert. Cut into 24 rectangles to serve. Store leftovers in refrigerator.