Prepare the Crust: Preheat your oven to 350°F (175°C). Line a 9x13-inch baking pan with parchment paper, leaving an overhang on the sides for easy removal. In a large mixing bowl, combine the sugar cookie mix, French vanilla pudding mix, melted butter, egg, brown sugar, and vanilla extract. Stir until a soft dough forms. Press the dough evenly into the bottom of the prepared pan to form the crust. Use your fingers or the back of a spoon to smooth it out.

Pre-Bake the Crust: Bake the crust for 10 minutes, then remove it from the oven and set aside to cool slightly while you prepare the filling.

Prepare the Cheesecake Filling: In a large mixing bowl, beat the cream cheese until smooth and fluffy. Add the egg, egg yolks, sour cream, granulated sugar, and vanilla extract. Mix until smooth and fully combined. Gently spread the cream cheese mixture over the pre-baked crust, smoothing the top with a spatula.

Add Toffee Bits and Bake: Sprinkle the toffee bits evenly over the top of the cream cheese filling. Return the pan to the oven and bake for 25-30 minutes, or until the edges are set and the center has a slight jiggle.

Cool and Chill: Allow the bars to cool at room temperature for 30 minutes, then transfer them to the refrigerator to chill for at least 3 hours or overnight. Chilling ensures the cheesecake layer sets perfectly.

Serve: Once fully chilled, lift the bars out of the pan using the parchment overhang. Slice into squares or rectangles and serve. Optional: Add a light dusting of sugar on top and torch with a kitchen blowtorch for an authentic "crème brûlée" caramelized finish.