Preheat oven to 350 degrees F (175 degrees C).

Melt the butterscotch chips and butter together in a microwave-safe glass or ceramic bowl in 30-second intervals, stirring after each melting, for 1 to 3 minutes (depending on your microwave). Stir butterscotch chip mixture and graham cracker crumbs together in a bowl. Press 1/2 the graham cracker crumb mixture into the bottom of a 9x13-inch baking dish.

Beat condensed milk, cream cheese, egg, and vanilla extract with an electric mixer until light and fluffy. Pour cream cheese mixture over the graham cracker crumb mixture in the baking dish. Spread remaining 1/2 graham cracker mixture on top of cream cheese mixture.

Bake in the preheated oven until set, 25 to 30 minutes. Chill completely before cutting.