Preheat the oven to 400°F (200°C).
Prepare the dough by mixing all-purpose flour and cold butter until crumbly.
Add cold water gradually and knead until the dough forms a ball. Chill for 15 minutes.
In a skillet over medium heat, cook ground beef until browned. Drain excess fat.
Add diced onions, bell peppers, tomato sauce, Italian seasoning, salt, and pepper to the beef. Simmer for 5 minutes and remove from heat.
Roll out dough on a floured surface to 1/8-inch thickness. Cut into 4 equal rectangles.
Divide the meat mixture and shredded mozzarella evenly onto half of each rectangle.
Fold the dough over to enclose the filling and seal edges by pressing with a fork.
Brush each hot pocket with beaten egg for a golden finish.
Place on a baking sheet lined with parchment paper and bake for 20-25 minutes until golden brown and crispy.
Let cool slightly before serving with a ramekin of marinara sauce.