Heat grill to 425F degrees with an indirect heat area.
Cut the pumpkin in half vertically. Using a spoon, scoop out the seeds and pulp. (Save the seeds to make smoked pumpkin seeds).
Brush the insides of the pumpkin with melted butter. In a small bowl, combine brown sugar and pumpkin pie spice. Sprinkle evenly over the inside of the pumpkin halves. Place on the grill over indirect heat cut side up. Grill with the lid closed for 1 hour.
While the pumpkin is cooking, beat together the cream cheese, sour cream, sugar, vanilla and salt for about 60 seconds, until it's smooth and creamy. Beat in the egg, until mixed through.
Reduce the grill heat to 350F degrees.
Spoon the cheesecake mixture into the pumpkins. Continue grilling with the lid closed for 1 hour or until the cheesecake is set.