Preheat the oven to 350°F. Line a 12-cup muffin pan with paper liners.

To prepare the cream cheese filling, beat together the cream cheese, sugar, and sour cream in a large bowl with an electric mixer on medium-high until smooth, approximately 1 to 2 minutes. Scrape down the sides of the bowl, as needed. Add the egg and continue to beat for an additional minute until fully incorporated. Transfer the mixture to a piping bag and set aside.

For the crumb topping, mix the brown sugar, cinnamon, and salt in a medium bowl. Drizzle in the melted butter and gently stir with a fork until combined. Add the flour and mix until small clumps form. Set the mixture aside at room temperature.

To make the pumpkin batter, whisk together the flour, baking soda, cinnamon, salt, and baking powder in a medium bowl. In a separate large bowl, whisk the pumpkin puree, sugar, oil, vanilla extract, and egg until the mixture is smooth. Gradually add the dry ingredients to the wet mixture, stirring until just combined.

Evenly divide the pumpkin batter between the prepared muffin cups, filling each about three-quarters full. Insert the piping bag's tip into the center of each muffin, about three-quarters of the way down, and squeeze approximately 2 tablespoons of cream cheese filling into the middle. After piping, the muffin cups should be nearly full. Gently press the crumb topping over each muffin.

Bake for approximately 30 minutes, or until the muffins have puffed up and the crumb topping is golden brown. A toothpick inserted near the edge of the muffin (avoiding the cream cheese) should come out clean. Allow the muffins to cool in the pan for 10 minutes before transferring them to a wire rack to cool completely.