Combine 1/4 cup flour, ice water, and 1 teaspoon vanilla, stirring with a whisk until well-blended; set aside. Combine 3/4 cup flour, 1/4 cup cocoa, sugar, and salt in a bowl; cut in shortening with a pastry blender or 2 knives until mixture resembles coarse meal. Add ice water mixture; toss with a fork until moist and crumbly (do not form a ball). Gently press mixture into a 4-inch circle on heavy-duty plastic wrap; cover with additional plastic wrap. Roll dough, still covered, into a 13-inch circle. Place dough in freezer 30 minutes or until plastic wrap can be easily removed.
Remove top sheet of plastic wrap; fit dough, uncovered side down, into a 10-inch round removable-bottom tart pan coated with cooking spray. Remove remaining sheet of plastic wrap. Fold edges under; flute. Pierce bottom and sides of dough with a fork; bake at 350° for 4 minutes. Cool on a wire rack. Place tart pan on a baking sheet; set aside.
Beat 1/2 cup cocoa and milk at medium speed of a mixer until blended. Add cheese; beat well. Add 2 teaspoons vanilla, egg, and egg white; beat just until smooth. Pour mixture into crust; bake at 350° for 25 minutes or until set. (Do not overbake.) Cool completely on a wire rack.
Spread whipped topping over tart; sprinkle with chopped chocolate.