For the dough: In a microwave-safe bowl, add the milk, butter, peanut butter and 80 milliliters (1/3 cup) water and microwave until the butter is melted, about 30 seconds. Allow to cool until lukewarm to the touch.
In the bowl of a stand mixer, add the all-purpose and bread flours, superfine sugar, salt and yeast and attach a dough hook. Turn the mixer to low speed and then slowly pour in the butter mixture. Add the egg and knead on medium speed for 2 minutes. Transfer the dough to a floured work surface. Knead with your hands until you have a smooth, soft and elastic dough. Put the dough into a lightly oiled bowl, cover and let stand for 15 minutes.
Transfer the dough to a floured work surface. Using a rolling pin, roll the dough into a rough rectangle about 17 inches wide and 12 inches tall. The dough should be about 1/4 inch thick. Use a pastry wheel or knife to trim off any rough edges and create an even rectangle.
Butter a 7-inch cake tin with a removable base and set aside.
For the filling: Spread the peanut butter on top of the dough to the edges, leaving a small (1/2-inch) border uncovered at the top and bottom of the dough (tip: if your peanut butter is slightly too stiff to spread, pop it in the microwave for 30 seconds). Repeat the process with the jam. Use a knife or pastry wheel to cut long strips of dough 2 inches wide. Roll each strip into a tight spiral and transfer to the prepared cake tin. Leave enough space between each for them to proof. Cover with a damp tea towel and allow to rise until the buns have doubled in size, 1 to 1 1/2 hours.
Preheat the oven to 170 degrees C/340 degrees F.
Bake until golden brown on top, about 25 minutes. If the buns are browning too quickly, simply cover them with aluminum foil.
For the icing: Meanwhile, add the powdered sugar and vanilla to a medium bowl. Add 1 tablespoon of milk and whisk the mixture together. Add more milk, 1 teaspoon at a time, until the icing is lump-free and thick but still runny.
Assembly: Remove the buns immediately from the cake tin, then spread the icing over top. Best served immediately.