Preheat the oven to 350ºF (180°C).
Butter or spray an 8-inch square pan. You can line it with a strip of parchment paper, covering the pan's bottom and two sides. The other two sides will remain unlined but greased. This will help you remove the bars from the pan.
Combine 1 and ½ cups flour, ½ cup light brown and ½ cup white sugar, ⅓ cup butter, and ½ teaspoon salt in a large bowl.
With an electric mixer, a pastry cutter, or your hands, mix well until crumbly.
Measure ⅔ cups of the mixture into a separate bowl o cup and set aside. This will be the crumble topping.
To the rest of the crumble add the 1 cup pumpkin puree, ⅓ cup sour cream, 2 tablespoons sugar (extra), 1 teaspoon pumpkin spice mix, ½ teaspoon baking powder, ¾ teaspoon baking soda, and ½ teaspoon vanilla. Mix to combine, beating on low speed only until mixed.
Add 2 eggs and beat until well incorporated. Don’t overmix.
Pour the mixture into the pan, spreading evenly.
Scatter the caramelized apples on top. Sprinkle the apples with the reserved crumble.
Bake for 50-60 minutes, or until golden and a cake tester or toothpick inserted in the middle comes out clean.
Let cool on a wire rack for 20 minutes. While still warm, run a smooth-bladed knife around the edges to loosen up any bit that might be stuck.
Remove from the pan lifting the whole block with the help of the paper. Let cool completely on the cooking rack.
Serve at room temperature or warm with ice cream or whipped cream and maybe a drizzle of caramel sauce.
Store leftovers covered, at room temperature for a day, and then refrigerate.