Using a standing mixer fitted with the hook attachment, combine the lukewarm water and yeast; let stand for 5 minutes. Add the flour and salt and mix on medium speed until the dough is smooth and wraps around the hook, about 5 minutes. Grease a large bowl with olive oil, transfer the dough to the bowl and let rise, covered, for 1 1/2 hours.
Preheat the oven to 375 degrees . In a medium skillet, cook the meat sauce over medium heat, stirring, until reduced to 1 1/2 cups, about 10 minutes; let cool.
Punch down the risen dough and transfer to a lightly floured work surface. Using a rolling pin, roll out to a 12-by-16-inch rectangle. Scatter the cheese on top, leaving a 1 1/2-inch border. Spoon the meat sauce over the cheese. Brush the border with some of the beaten egg and roll the dough up loosely, beginning at the short end. Transfer to a parchment-paper-lined baking sheet, brush with more egg and cut 3 vents on top. Bake until golden-brown, about 40 minutes.