Make the donut dough: In the bowl of a stand mixer fitted with a dough hook, combine the flour, yeast, sugar, warm milk, pumpkin puree, salt, and melted butter.
Mix on medium speed until dough is smooth and elastic and no longer sticks to the side of the bowl.
Place dough on a lightly floured surface and shape into a ball.
Place dough in a large, lightly oiled bowl and cover with plastic wrap.
Let dough rise in a warm place for 1-2 hours (or overnight in the fridge for up to 12 hours) until it has doubled in size.
While the dough rises, make the pumpkin pie filling: In a large heatproof bowl, whisk together the egg yolks and sugar. Set aside.
In a small bowl, whisk together the flour and water until smooth. Set aside.
Add pumpkin puree, vanilla extract, and evaporated milk to a medium saucepan and mix together until smooth. Place on the stove over medium heat until mixture is steaming.
Remove the pan from heat and pour the hot pumpkin mixture into the egg yolk mixture, whisking constantly, until smooth.
Return the pumpkin mixture to the saucepan along with the flour mixture and whisk until combined. Heat over medium-low heat, whisking often, until mixture is thickened to the point where you can draw a clean line through it and it holds its shape.
Pour the custard into a bowl and cover with plastic wrap, pressing the wrap into the top of the custard so it won't form a skin. Refrigerate until ready to use.
Once the donut dough has risen, place the dough on a lightly floured surface. Roll into a large rectangle about ½ inch thick.
Use a 2 ½ inch round cookie cutter to cut out circles in the dough, re-rolling the scraps until you have enough donuts.
Place the dough circles on a baking sheet and cover with plastic wrap.
Fill a large, heavy-bottomed pot or Dutch oven with at least 2 inches vegetable oil. Place over medium heat until it reaches 360 F.
Add about 4 donuts at a time to the oil, frying for about 2 minutes per side until golden.
Drain donuts on a paper towel-lined plate. Repeat this process until all the donuts are fried.
In a small bowl, whisk together the granulated sugar and pumpkin pie spice.
While they are still warm, roll the donuts in the sugar mixture to coat. Place on a wire rack set over a baking sheet.
Use a chopstick to make holes on both sides of each donut.
Add pumpkin custard to a piping bag fitted with a large tip. Fill the donuts with the custard.