Heat oven to 350°F. Line 12-inch pizza pan with foil; grease foil with shortening. Cut cookie dough into 1/4-inch slices; arrange evenly in pan. With floured fingers, press slices to form crust.
Bake 16 to 20 minutes or until deep golden brown. Cool completely, about 30 minutes.
In small bowl, beat cream cheese, sugar and vanilla with electric mixer on medium speed until fluffy. Use foil to lift crust from pan. Carefully remove foil from crust; place crust on serving platter or tray. Spread cream cheese mixture over crust.
In a small resealable food-storage plastic bag, place chocolate chips and oil; seal bag. Place bag in bowl of very hot water. Knead bag several times to melt chips. Cut small hole in 1 corner of bag. Working from center to outer edge, pipe chocolate to form spiral design over top of cream cheese mixture. Working from center of spiral to outer edge, draw tip of knife lightly through spiral to form web design.
Refrigerate until chilled, at least 1 hour. To serve, cut into wedges or squares. Cover and refrigerate any remaining pizza.