In a medium saucepan over medium heat, combine the water and butter. Once the butter has melted, stir in the pumpkin and salt until fully incorporated.
Add the flour and mix until the dough starts to form a ball and pulls away from the sides of the pan.
Keep stirring until the mixture is cohesive—it will be sticky, but that's normal. This process should take about 30 seconds.
Transfer the dough to a bowl (or the bowl of a stand mixer with the paddle attachment) and mix briefly to release some of the steam. Let the dough cool for about 10 minutes.
Next, add one egg at a time, along with the vanilla, mixing thoroughly between each addition. Continue mixing until the dough is smooth and glossy. Transfer the dough to a piping bag fitted with a star tip.
In a shallow bowl, mix together the sugar, pumpkin pie spice, and a pinch of salt. This will be used to coat the churros after frying.
Heat about a quart of oil in a deep pan to 180°C. Once the oil reaches temperature, reduce the heat to medium-low to help maintain it. Pipe the churro dough directly into the hot oil, using clean kitchen scissors to cut the dough as it’s piped.
Fry the churros for about 1 minute on each side, or until golden brown. Transfer them to paper towels to drain the excess oil, then immediately roll them in the spiced sugar mixture, ensuring they are evenly coated. Repeat with the remaining dough.