Combine the yeast with 2 ¼ cups of warm water in the bowl of a stand mixer.
Add the olive oil, flour, and salt to the water mixture and mix with a spatula or wooden spoon until a shaggy dough forms.
Mix with a dough hook in your stand mixer for 5 minutes, or until the dough is smooth and elastic.
Cover and let rise for 2 hours, or until doubled in size.
Add the sausage and the olive oil to a large saute pan and heat on medium heat.
Add the garlic, peppers, and onions to the pan and cook, stirring and breaking up the meat, for 5 minutes, or until the meat is browned and the vegetables are softened.
Add the dried basil, salt, and pepper, remove from the heat, and set aside.
Remove the cover from the dough and place the dough ball on a flour-dusted surface.
Roll the dough out into a rough rectangle.
Cut the rectangle in half and roll out each half into a 9x13-inch rectangle.
Divide the filling in half — you will use half of the filling for each sausage bread — then place some of the filling along the center of one rectangle and top with 5 slices of the provolone cheese.
Place the remainder of this half of the filling over the cheese, then top with 5 slices of the mozzarella cheese and 2 tablespoons of the Parmesan cheese.
Wrap the dough over the filling and seal the edges and center together.
Turn the dough so that the seam is facing down on the parchment paper.
Beat the egg in a small bowl.
Place the sausage bread on a baking sheet and brush the top with the egg wash.
Sprinkle the top of the bread with some of the grated Parmesan cheese.
Score the top of the bread with three diagonal cuts. Repeat steps 11–18 with the second dough.
Preheat the oven to 375 F.
Bake for 40 minutes, or until golden brown and cooked through.
Allow the bread to cool slightly, then slice and serve.