In a big bowl, combine egg, instant dry yeast, warm milk, granulated sugar, vanilla extract, and salt. Mix thoroughly until the sugar dissolves with a mixer.
Add all-purpose flour, knead the dough for 15 minutes until the dough is smooth, and elastic then add butter and knead again until the dough is not too tacky, about 15 minutes with a mixer.
Cover it with a wet cloth or cling wrap so the dough won't dry out. Proof the dough for 45 minutes or until double the size.
Cut the parchment paper into 12x6 cm. Sprinkle flour on top of the parchment paper to prevent the dough from sticking when it will be fried.
Once the dough has risen, portion the dough of the bomboloni 40 g each.
Make a smooth ball shape, then flatten it a bit, and then put it on parchment paper, and proof it again for 15 minutes.
Heat oil in a pan on medium heat. Fry the bomboloni at 150°C or 300°F until both sides golden brown.
Rest them in the cooling rack only for a couple of seconds, then roll the bomboloni immediately in granulated sugar, be sure to cover them well.
To fill the bomboloni, insert a scissor into the side of a bomboloni and then open the scissors when it's inside the bomboloni to make a small space.
Fill a piping bag with chocolate hazelnut jam and pipe it into the bomboloni.