Preheat the oven to 375°F (190°C).
Heat olive oil in a large skillet over medium heat.
Add ground beef and cook, breaking it up with a spoon, until browned and cooked through, about 6-8 minutes.
Add chopped onion and minced garlic to the skillet and cook until softened, about 3 minutes.
Stir in sliced mushrooms, diced green and red bell peppers, and sliced black olives. Cook for 5 minutes until vegetables are tender.
Add dried oregano, dried basil, dried thyme, crushed red pepper flakes, salt, and black pepper. Stir to combine.
Pour in crushed tomatoes and tomato sauce. Simmer the mixture for 10 minutes, stirring occasionally, until slightly thickened.
Remove the skillet from heat and stir in shredded mozzarella and provolone cheeses until melted and well combined.
Transfer the beef and vegetable mixture into a 9-inch deep dish pie pan, spreading it evenly.
Place the refrigerated pie crust over the filling, tucking the edges inside the pie pan. Trim any excess dough and crimp the edges to seal.
Brush the top of the pie crust with the beaten egg to achieve a golden brown finish.
Cut a few small slits in the top crust to allow steam to escape during baking.
Bake the deep dish pizza pot pie in the preheated oven for 35-40 minutes, or until the crust is golden brown and the filling is bubbly.
Remove from the oven and let it rest for 10 minutes before slicing and serving.