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Preheat the oven to 375°F. Melt 2 tablespoons of butter and spread it evenly in a 9×13-inch baking dish, sheet pan, or another pan with sides.
Place the rolls over the butter. Cut an X into the rolls about ⅔s down but not completely through the bottom of the rolls.
Stir together the ⅓ cup melted butter and pumpkin pie spice in a measuring cup. Pour evenly over the rolls and in the Xs you cut.
Stir the pumpkin, pumpkin spice, egg, and sugar together.
Now, you'll fill the rolls with the pumpkin mixture. (It's easier if you use a piping bag with a large tip on it to pipe the pumpkin into the rolls. If you don't have a piping bag, you can up a zip-top bag and snip the end. You'll need to work with it to get the pumpkin mixture into the roll and not just on top. Or, you can spoon the pumpkin into the rolls. It may take a little working with the first time you do it. I found the piping bag and tip the easiest.)
Bake the rolls, uncovered at 375°F for 12 to 15 minutes. Everything is already cooked except the egg and they just need to be at 145°F for 15 seconds. (You can test with an instant-read thermometer if you prefer.) Therefore, after 12 minutes it really depends on how crusty you want the rolls to get.
While they're cooking whisk the sifted powdered sugar and maple syrup together until smooth.
Allow the rolls to cool just a couple of minutes before serving pouring the glaze over them.
Serve warm. Store any leftovers in an airtight container in the refrigerator. They'll keep 3 to 4 days.