Heat the olive oil in a large Dutch oven or heavy-bottomed pot over medium-high heat. Season the beef cubes generously with salt and black pepper, then brown them in two batches until a dark crust forms on all sides, about 5 to 7 minutes per batch. Transfer the browned beef to a plate and set aside.

Reduce the heat to medium and add the chopped onion and diced carrots to the pot. Sauté until the onions are softened and slightly translucent, about 5 minutes, using a wooden spoon to scrape up the flavorful browned bits from the bottom of the pot.

Sprinkle the flour over the softened vegetables and stir constantly for 1 minute to cook off the raw flour taste. Gradually pour in the beef broth, stirring continuously to prevent any lumps from forming.

Return the browned beef and any accumulated resting juices to the pot. Stir in the Dijon mustard, whole grain mustard, and dried thyme. Bring the mixture to a gentle boil, then reduce the heat to low, cover, and simmer for 60 to 70 minutes, or until the beef is fork-tender.

Preheat your oven to 400°F (200°C). Pour the thickened beef filling into a 9-inch deep-dish pie pan or a medium baking dish. Allow the filling to cool slightly for about 10 minutes so the pastry doesn't melt upon contact.

Unroll the thawed puff pastry sheet and carefully drape it over the pie dish. Trim any excess overhanging pastry and crimp the edges to seal it against the dish. Cut three small slits in the center of the pastry to allow steam to escape, then brush the entire surface evenly with the beaten egg.

Bake in the preheated oven for 20 to 25 minutes, until the puff pastry is fully puffed and deep golden brown. Let the pie rest for 10 minutes before slicing and serving alongside buttered green peas or mashed potatoes.