Season the beef cubes with salt and black pepper.
Heat the olive oil in a large skillet over medium-high heat. Add the beef cubes in batches and brown on all sides, about 3-4 minutes per batch. Transfer browned beef to the slow cooker.
In the same skillet, add the diced onion and cook for 3-4 minutes until softened. Add the minced garlic and cook for 1 minute until fragrant.
Sprinkle the flour over the onions and garlic, stirring constantly for 1-2 minutes to cook the flour.
Slowly whisk in the beef broth, scraping up any browned bits from the skillet. Stir in the tomato paste until fully combined.
Pour the mixture over the beef in the slow cooker. Add the sliced carrots, celery, dried thyme, dried rosemary, and bay leaf. Stir gently to combine.
Cover the slow cooker and cook on low for 6 hours, or until the beef is tender and the vegetables are cooked through.
About 30 minutes before serving, remove the bay leaf and stir in the frozen peas and corn. Cover and continue cooking.
Preheat the oven to 400°F (200°C). Transfer the beef and vegetable mixture from the slow cooker to an oven-safe baking dish, spreading it evenly.
Roll out the puff pastry sheet to fit the top of the baking dish. Place the pastry over the beef mixture, trimming any excess edges. Press the edges to seal and cut a few small slits in the top to allow steam to escape.
Brush the puff pastry with the beaten egg for a golden finish.
Bake in the preheated oven for 25-30 minutes, or until the puff pastry is puffed and golden brown.
Remove from the oven and let rest for 5 minutes before serving.