Place the yeast and sugar in the bowl and pour warm milk on top of it. Whisk gently to combine, then loosely cover with a clean kitchen towel and allow to sit for 5-10 minutes. The mixture will be frothy after 5-10 minutes.
In a large mixing bowl add the butter, egg, flour, salt and yeast mixture. Form a soft dough out it. Using lightly floured hands, knead it for 1 minute. If the dough is too sticky to handle, add 1-3 more tbsp of flour, but you want a very soft dough. Shape into a ball.
Place the dough in a greased bowl and cover with plastic wrap or aluminum foil. Place in a slightly warm environment to rise until doubled in size, around 60-90 minutes.
As the dough rises, prepare the filling and grease a 9×5 inch loaf pan.
Punch down the dough to release the air. Place dough on a lightly floured work surface. Roll it into large rectangle.
Brush olive oil followed by the pesto sauce. Then, sprinkle the shredded cheese.
Using a sharp knife, slice the rolled dough into roughly 5 equal parts. Stack the slices on one top of each other and then slice the stacked dough into 5 roughly equal size squares/rectangles.
Take a 9x5 loaf pan and carefully place the sliced squares/rectangles one after the other.
Cover with plastic wrap or aluminum foil and allow to rise once again in a slightly warm environment until puffy, about 45 minutes.
Adjust the oven rack to the lower third position then preheat oven to 350°F (177°C). Bake until golden brown, about 50 minutes.
Remove from the oven and place the pan on a wire rack. If desired, brush with melted butter.
Cool for 10 minutes in the pan, then remove from the pan and serve warm.