Dissolve the yeast in the lukewarm water. Add one egg along with the salt and sugar. Whisk until mixed. Add the flour a little at a time. The dough should feel a little dry. Knead for 7-8 min. Add the butter in cubes. Knead until all the butter has been distributed throughout the dough. Let the dough rise covered with a cloth until double in size.
Roll the dough out as thinly as possible. Try to form it into a square if you can. Distribute the grated cheese across one half of the dough, then cover with the prosciutto and basil. Fold the other half of the dough over to cover the filling. Cut the dough lengthwise into 2cm strips. Twist the dough around itself until you have a long twisted sausage, then twist the sausage around itself. Place the swirls on a baking sheet lined with parchment paper and let rise for half an hour. Brush with beaten egg and grate a little extra cheese over them if you like.
Bake at 200°C for approx. 15-18 min., depending on their size. They should be slightly golden and sound hollow when tapped on the bottom.