In a bowl of a stand mixer combine warm milk(not hot!), instant yeast, and a tablespoon of the total amount of sugar. Let them sit for about 10-15 minutes giving time for the yeast to activate.
Meanwhile melt the butter.
When the yeast has foamed up add the cooled melted butter, the rest of the sugar, eggs, and pumpkin puree. Mix everything on medium speed with the dough hook attachment.
After everything is combined, turn the speed on low and add the salt, then gradually add the flour. Mix for about 5-8 minutes, till the dough is smooth and starts to come off the sides of the bowl. It won't be completely removed from the bottom and a bit sticky, but that's ok.
Oil another bowl and transfer the dough in it. Shape it into a ball and let it rest covered with a towel for an hour or longer, depending on the room temperature.
When the dough has doubled its size, transfer it to a flat surface. Divide the dough into about the same size pieces. I use a kitchen scale to measure them. Each piece comes to about 105g. Roll them into balls.
Take one ball at a time and flatten it using your hands or use a rolling pin. It should be about 5"(13cm) in diameter. Then take a tablespoon of the pumpkin jam and place it in the center of the dough. Seal the ends by pinching them to the center. Then using the palm of your hand roll the ball with the seam side down, to create a smooth bun-like shape. Place the bun with the seam side down on parchment paper. Repeat the same with the rest of the dough.
Make sure you leave about 2"(5cm) space between each bun. Cover the buns with a towel and let them rest for another hour.
About 30 minutes before baking turn on the oven to 350F(180C). When ready to bake, make the egg wash by whisking the egg with a tablespoon of water or milk. Brush each bun with the egg wash using a pastry brush. Sprinkle more sugar on top.
Bake on the middle rack for about 15 minutes, depending on your oven. They should look golden brown on top.
Remove from the oven and transfer to a cooling rack. Let them cool down a bit before enjoying.