Combine flour, yeast, salt and sugar in a stand mixer with the dough hook. Add warm milk, butter and egg.
Knead on medium-low 8 minutes until soft and elastic. Place in an oiled bowl, cover, prove 60 minutes until doubled.
Roll out into a 30x40cm rectangle on a floured bench. Spread vegemite thinly over the surface, leaving a 2cm edge. Sprinkle with most of the cheese.
Roll up tightly along the long edge. Cut into 12 equal slices. Place cut-side-up on a lined tray, leaving 2cm gaps.
Cover loosely. Prove 30 minutes.
Preheat oven to 200°C fan. Brush scrolls with beaten egg. Top with remaining cheese.
Bake 18-22 minutes until deep golden and the cheese is bubbling.
Cool 10 minutes on the tray. Eat warm.