Preheat the oven to 350°F (177°C). Brush an 8 by 8 inch pan with oil and line it with parchment paper, leaving an overhang on 2 sides.
In a large bowl, beat the ¾ cup of melted butter with the 1 cup of sugar and the 1 teaspoon of vanilla. Add the 2 eggs and beat until light and fluffy.
In a separate bowl, whisk together the ½ cup of flour, the ½ cup of cocoa powder, the ¼ teaspoon of salt and the 2 teaspoons of cinnamon. Add to the butter mixture and mix until smooth.
In a third bowl, beat the 8 ounces of softened cream cheese until smooth. Add the 1 egg, the ⅓ cup of sugar, the ¼ cup of flour, the ½ cup of canned pumpkin, the 1 teaspoon of vanilla, the ½ teaspoon of cinnamon and the 1 teaspoon of pumpkin spice. Beat until smooth.
Spread about ⅔ of the brownie batter evenly in the prepared pan, into the corners. Spoon the cheesecake batter over it and spread gently. Dollop the remaining brownie batter over the top, then drag a knife or skewer through the top 2 layers to swirl, 4 or 5 passes. Do not swirl down into the bottom layer.
Bake for 40 to 50 minutes, until the center is set.
Cool completely in the pan on a wire rack, about 1 hour 30 minutes.
Lift out by the parchment or cut in the pan, into 16 squares. For the cleanest edges, chill 1 hour before cutting.
To serve warm, microwave a square for 15 seconds, or warm on a tray in a 300°F oven for 5 minutes. Top with ice cream.