Heat the olive oil in a large oven-safe pot or Dutch oven over medium-high heat. Add the beef cubes in batches, searing until browned on all sides, about 5 to 7 minutes per batch, then remove the beef and set aside.

Lower the heat to medium and add the chopped onion and carrots to the pot. Saute until the vegetables are softened and lightly browned, about 5 minutes, then stir in the minced garlic and cook for 30 seconds until fragrant.

Sprinkle the all-purpose flour over the vegetables, stirring continuously for 1 minute to cook out the raw flour taste. Gradually pour in the beef broth, scraping up any browned bits from the bottom of the pot.

Return the browned beef to the pot along with the whole-grain mustard, Dijon mustard, fresh thyme, salt, and black pepper. Bring the mixture to a simmer, then cover, reduce heat to low, and let it cook for 1.5 hours, or until the beef is very tender and the gravy has thickened.

Preheat your oven to 400 degrees Fahrenheit (200 degrees Celsius). Transfer the hot beef filling into a 9-inch deep pie dish or baking dish, spreading it out into an even layer.

Unroll the thawed puff pastry sheet and drape it over the pie dish, trimming any excess edges and crimping them to seal against the rim. Brush the beaten egg evenly over the pastry, then cut three small slits in the center to let steam escape.

Bake in the preheated oven for 20 to 25 minutes, or until the pastry is puffed and deeply golden brown. Let the pie rest for 10 minutes before serving with a side of steamed green beans or mashed potatoes.