Grease the inside of a large bowl with butter or oil, and line 2 large baking sheets with parchment paper. Set aside.
Heat water in a small bowl or cup for 10-20 seconds until it reaches 105°F-110°F. Mix in yeast, 1 tablespoon sugar and loosely cover bowl with plastic wrap. Let it sit for 5 minutes. If the mixture is bubbly or frothy, that means the yeast is alive and thriving. Continue to the next step. If the mixture hasn't really changed at all, throw the mixture out and start again with a new packet of yeast. (I recommend using a thermometer to make sure your water isn't too hot, otherwise it could kill the yeast.)
In the bowl of a stand mixer (or a regular large bowl), add the flour, remaining sugar, pumpkin spice, and salt. Mix together to combine.
Add the melted butter and 1 egg. Mix together again to lightly combine.
Add in the yeast mixture and knead in the stand mixture using the dough hook for 6-8 minutes (or by hand), until the dough is smooth and elastic. The dough should be slightly tacky, but not completely sticky. If it is sticky, add in more flour 1 tablespoon at a time until the dough is no longer completely sticky.
Place the dough in the prepared bowl, cover loosely with a sheet of plastic wrap, and let it rise in a warm place for 2 hours, or until the dough has about doubled in size. (I like to let mine rise on top of the fridge.)
While the dough is rising, make the filling: Add the pumpkin puree, dark brown sugar, vanilla extract, and pumpkin spice to a small saucepan over medium-high heat. Stir together to combine and cook for 3 minutes, stirring occasionally.
Remove from heat, transfer to a bowl, and set aside until ready to use.
Preheat the oven to 350°F. Divide the dough into 16 equal pieces and shape into balls.
On a lightly floured surface, roll each ball into a 6-7 inch circle. Place a heaping tablespoon of the pumpkin filling onto each flattened dough circle. Fold one side over and pinch the sides together using your index finger until the empanada is fully sealed (see photos in blog post for example). You can also press the edges together using the tines of a fork instead of your fingers. As long as the empanadas are sealed, you're good to go.
Transfer the empanadas to the prepared baking sheets. In a small bowl, whisk the remaining 1 egg. Using a pastry brush, brush the tops of the empanadas with the egg wash.
Place in the oven and bake for 23-25 minutes, until the tops are golden brown.
Transfer to a wire rack and let sit for 30 minutes to cool before serving.