Step 1: Preheat oven to 350°F (175°C). Grease and flour a 9-inch springform pan. Line the bottom with parchment paper.
Step 2: In a medium saucepan, melt 1/2 cup butter. Add 1 cup sugar and stir. Remove from heat.
Step 3: Whisk in 2 eggs, one at a time, then 1 teaspoon vanilla extract.
Step 4: Sift together flour, cocoa powder, baking powder, and salt. Gradually add to wet ingredients, mixing until just combined.
Step 5: Pour brownie batter into the prepared pan. Bake for 20-25 minutes, then cool completely in the pan.
Step 6: Reduce oven temperature to 325°F (160°C).
Step 7: In a large bowl, beat softened cream cheese until smooth, about 2-3 minutes. Scrape down sides.
Step 8: Add 1 cup sugar and beat until just combined. Beat in 1 teaspoon vanilla extract.
Step 9: Add 3 eggs, one at a time, beating on low speed until just combined after each.
Step 10: Pour cheesecake batter over the cooled brownie base.
Step 11: Prepare a water bath: Wrap the springform pan tightly with foil. Place it in a larger roasting pan and fill with hot water halfway up the sides of the springform pan.
Step 12: Bake for 55-65 minutes, or until edges are set with a slight wobble in the center. Turn off the oven, crack the door, and cool in oven for 1 hour.
Step 13: Remove from water bath, cool completely on a wire rack, then cover and refrigerate for at least 6-8 hours or overnight.
Step 14: Gently remove the chilled cheesecake from the springform pan.
Step 15: Spread 1 cup caramel sauce evenly over the top.
Step 16: Sprinkle 1 cup toasted and chopped pecans over the caramel.
Step 17: Melt 1/2 cup chocolate chips with 1 tablespoon butter. Drizzle over the pecans and caramel.
Step 18: Refrigerate for another 30 minutes for the topping to set before slicing and serving.