Combine the dough ingredients in the bowl of your stand mixer, and use the dough hook to stir on low till combined then turn the speed up to 6 and knead for 5 minutes.
If the dough feels sticky add another tbsp of flour and knead for another 3-4 minutes at which point if you pick up a piece of dough and stretch it it will be nice and elastic and not break.
Break a small piece off this dough, about golf ball sized, and wrap it around the frozen butter pieces and put them back in the freezer. The dough should stretch in a very thin film around the butter - a too thick dough won't bake through or give you a doughy center.
Leave the remaining dough to rise for an hour in a lightly oiled bowl covered with a kitchen towel inside a draft free space (I put it inside my oven with the oven off).
After an hour give your rise dough a quick turn by hand and divide it into six equal pieces. Flatten each piece out into a circular shape, place the pat of frozen butter inside and form a circle making sure the bottom is well sealed.
Leave them to rise on a lined baking tray for 45 minutes. Gently cover the top with a kitchen towel or a piece of plastic wrap. The dough balls will expand some but this step is important for the texture of the buns!
Coffee Topping:
Beat the icing sugar and butter together, first on low for the icing sugar to incorporate then on higher speed. Best to do this with a hand mixer or a whisk
Now add the egg and whip then the vanilla and coffee. The mix may look curdled and that's fine.
Mix in the flour until just combined then put your coffee topping in an icing bag or use a ziplock instead of one.
Preheat the oven to 350F
Snip a 1 cm piece off the end of the icing bag and pipe the coffee topping on in swirls over the coffee buns, making sure to cover the surface.
Bake for 17 minutes - the coffee buns should be a medium brown and well domed. Dig in!