Preheat oven to 350 degrees F (175 degrees C). Spray a 9x13-inch baking dish with cooking spray.
Heat butterscotch chips and butter together in a saucepan, stirring frequently, over low heat until melted, about 5 minutes. Remove saucepan from heat and stir graham cracker crumbs into butterscotch mixture. Measure 2/3 cup graham cracker-butterscotch mixture and set aside; press remaining mixture into the prepared baking dish to form a crust.
Beat cream cheese in a bowl using an electric mixer until light and fluffy. Add sweetened condensed milk, egg, and vanilla extract; mix well. Pour cream cheese mixture over crust. Sprinkle reserved graham cracker-butterscotch mixture over cream cheese mixture layer.
Bake in the preheated oven until a knife inserted in the center comes out clean, 25 to 30 minutes.
