Preheat the oven to 350°F (175°C). Line a 9x13-inch metal baking pan with parchment paper, leaving overhang on two sides. Lightly grease the exposed ends.
Beat the softened butter, brown sugar, and granulated sugar for 2 to 3 minutes, until creamy and slightly lighter in color.
Mix in the egg, egg yolk, and 2 teaspoons vanilla. Whisk the flour, baking soda, and salt separately, then add to the butter mixture. Mix just until combined and fold in the chocolate chips.
Reserve one-third of the cookie dough. Press the remaining dough evenly into the prepared pan and bake for 8 minutes. The crust should still look pale and underdone.
Beat the softened cream cheese and 1/2 cup sugar until smooth. Mix in the sour cream, 1 teaspoon vanilla, and salt. Add the egg last and mix on low only until combined.
Spread the cheesecake filling evenly over the warm cookie crust. Crumble the reserved cookie dough over the filling, leaving small spaces where the cheesecake remains visible.
Bake for 28 to 32 minutes, until the cookie topping is lightly golden and the cheesecake is set around the edges with a slight wobble in the center.
Cool completely in the pan on a wire rack. Cover and refrigerate for at least 3 hours or overnight.
Lift the chilled slab from the pan using the parchment. Cut into 20 bars with a sharp knife, wiping the blade clean between cuts.