To make the dough, in the bowl of your stand mixer, combine the warm milk, melted butter, yeast and sugar. Let sit for about 10 to 15 minutes until the yeast starts to foam up a bit. Add in the egg, 3 1/2 cups flour and salt, and mix with the dough hook attachment until the dough begins to come together. Add in as much of the remaining flour, 1/4 cup at a time, as needed to create a soft and somewhat sticky dough. It should come away from the sides of the bowl and just slightly stick to the bottom. Allow the dough to knead on medium-high until soft, about 7 minutes. Remove the dough and transfer it to a lightly oiled bowl, turning it over to coat both side. Cover loosely with plastic wrap and a damp kitchen towel. Place the bowl in a warm spot to double in size, about 1 hour.
Transfer the dough to well floured work surface and roll out to about 1/2-inch thick. Cut into circle using a 3-inch round cutter. Place on a baking sheet lined with parchment paper. Cover with a damp kitchen towel and allow to rest while the oil heats up.
Fill a large heavy dutch oven about halfway with oil. Attach a candy or deep-frying thermometer and bring the oil to 350 degrees F. Once hot, fry the doughnuts in batches, for about 2 to 3 minutes on the first side. Flip over and continue to fry for another 1 to 2 minutes or until golden brown. Drain and transfer to a wire rack set over a baking sheet. Continue frying the rest of the doughnuts in the same manner.
Once all the doughnuts have been fried, give the jam a good stir to loosen it up. Transfer to a disposable piping bag, fitting with a small metal circle tip. Fill each doughnut with about 1 teaspoon (or more depending on how big your doughnuts are) of jam by poking a hole on the side of each with a sharp pairing knife and then piping the jelly into the doughnut. Return to the wire rack.
To make the glaze, whisk together the powdered sugar, peanut butter, vanilla, salt and as much milk as needed to create a smooth thin pourable glaze. It should be thin enough to drizzle. If it's too thin add a bit more powdered sugar and stir. If it's too thick, add more milk.
Dunk one side of each doughnut into the glaze and return to the pan. Allow the glaze to set before eating. Doughnuts are best eaten the day you make them, but leftovers can be stored in an airtight container or wrapped in plastic and kept for up to 2 days. Enjoy!
