In a large pot or Dutch oven, brown the beef chuck over medium-high heat. Remove beef and set aside.
Add onion and garlic to the pot and cook until softened, about 5 minutes.
Return the beef to the pot. Add potatoes, carrots, thyme, rosemary, and beef broth. Bring to a simmer, then reduce heat and cook, covered, for 1.5-2 hours, or until beef is tender.
Season with salt and pepper to taste.
Spoon the beef mixture into a baking dish.
Top with the puff pastry, crimping the edges to seal.
Brush the top with the beaten egg.
Bake for 25-30 minutes, or until the pastry is golden brown.
Notes
Serve hot and enjoy this hearty and flavorful Irish pie!