Preheat oven to 325°F (160°C). Butter a 9-inch square baking pan. Line the pan with parchment paper, leaving enough to extend over the sides. Butter the parchment paper.
Combine the graham cracker crumbs and butter until the crumbs are moistened. Stir in the chocolate chips.
Press the crust mixture into the bottom of the pan and 1 inch up the sides. Bake for 6 minutes. Set the pan on a wire rack to cool.
To make the cookie dough:
Using an electric mixer on medium speed, beat the butter, brown sugar, sugar, salt, and vanilla extract until smooth.
Decrease mixer speed to low and add flour. Mix just until incorporated. Stir in the chocolate chips. Set aside.
To make the filling:
Using an electric mixer on medium speed, beat the cream cheese and sugar just until smooth. Add the egg and vanilla extract, beating just until blended.
Pour the batter into the partially baked crust. Drop the cookie dough by teaspoonfuls over the top of the filling.
Bake 30 minutes, or until set. Transfer to wire rack.
To make the chocolate topping:
Melt the chocolate chips in the microwave or in a double boiler. Add butter if necessary to make the chocolate smooth.
Drizzle over the top of the bars. Cool the bars completely in the pan (about an hour). Then refrigerate for at least 4 hours.
Using the edges of the parchment paper, remove the bars from the pan. Cut into bars and serve.
Notes
Recipe slightly adapted from The Essential Chocolate Chip Cookbook.
Store in an airtight container in the refrigerator up to 5 days.