Preheat the oven to 375°F (190°C).
Heat olive oil in a large skillet over medium heat. Add the diced onion and cook until translucent, about 3 minutes.
Add the minced garlic and cook for 1 minute until fragrant.
Add the ground beef to the skillet and cook, breaking it up with a spoon, until browned and cooked through, about 6-8 minutes. Drain excess fat if necessary.
Stir in the sliced mushrooms, diced green and red bell peppers, dried oregano, dried basil, salt, and black pepper. Cook for 5 minutes until vegetables are tender.
Add the crushed tomatoes, tomato sauce, sugar, and red pepper flakes. Stir well and simmer for 10 minutes to let the flavors meld. Remove from heat.
Roll out the refrigerated pie crust and press it into a 9-inch deep-dish pie pan, covering the bottom and sides evenly.
Pour half of the tomato and beef mixture into the pie crust.
Sprinkle 3/4 cup of shredded mozzarella cheese evenly over the filling.
Add the remaining tomato and beef mixture on top of the cheese layer.
Sprinkle the remaining 3/4 cup mozzarella cheese and the Parmesan cheese evenly over the top.
Roll out the refrigerated pizza dough to cover the top of the pie. Place it over the filling and pinch the edges to seal the crusts together.
Brush the top crust with the beaten egg to create a golden finish.
Cut 3-4 small slits in the top crust to allow steam to escape during baking.
Bake in the preheated oven for 40-45 minutes, or until the crust is golden brown and the filling is bubbly.
Remove from the oven and let the pot pie rest for 10 minutes before slicing and serving.