Remove the egg, sour cream, buttermilk, and cream cheese from the refrigerator about 2 hours before baking so they can come to room temperature.
Preheat the oven to 425°F (220°C). Line muffin pans with paper liners, spacing the liners apart by using only 6 liners in each 12-cup muffin pan.
Spread the pumpkin puree onto a plate. Cover it with paper towels and gently blot the puree 3–4 times to remove excess moisture.
In a large bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, pumpkin pie spice, and nutmeg.
In a separate bowl, stir together the blotted pumpkin puree, melted and cooled butter, brown sugar, egg, vanilla extract, sour cream, and buttermilk until smooth.
Pour the wet mixture into the dry ingredients. Gently fold with a rubber spatula just until combined. Avoid overmixing; the batter should be thick with a few small streaks or lumps remaining.
Divide the batter between the prepared liners, mounding it almost to the brim. Sprinkle the tops with coarse sugar.
Beat the softened cream cheese with the granulated sugar, flour, milk, and vanilla extract until smooth and creamy.
Transfer the cream cheese mixture to a piping bag. Insert the tip about one-third of the way into each muffin and gently squeeze a dollop of filling into the center.
Bake the pumpkin cream cheese muffins at 425°F (220°C) for 8 minutes. Without opening the oven, reduce the temperature to 350°F (175°C) and continue baking for 6–8 minutes, or until a toothpick inserted into the muffin portion away from the cream cheese comes out clean.
Allow the muffins to rest in the hot pan for 10 minutes. Transfer them to a wire rack and cool completely before serving.