Heat olive oil in a large Dutch oven or heavy-bottomed pot over medium-high heat. Add the beef cubes and sear until browned on all sides, about 5 to 7 minutes.

Add the chopped onion and minced garlic to the pot with the beef. Cook until the onions are softened and translucent, which should take about 4 minutes.

Sprinkle the flour over the beef and onions, stirring constantly for 1 minute to coat the meat and cook off the raw flour taste.

Gradually pour in the beef broth, stirring well to scrape up any browned bits from the bottom of the pot. Stir in the whole grain mustard, Dijon mustard, thyme, salt, and pepper.

Bring the mixture to a boil, then reduce the heat to low, cover, and simmer for 1 hour. The beef should be very tender and the gravy nicely thickened. Remove from heat and let it cool slightly.

Preheat your oven to 400°F (200°C). Transfer the slightly cooled beef filling into a 9-inch deep pie dish.

Unroll the puff pastry and drape it evenly over the pie dish, crimping the edges to seal it against the rim. Brush the top evenly with the beaten egg and cut three small slits in the center to allow steam to escape.

Bake for 25 to 30 minutes until the pastry is puffed and deeply golden brown. Let the pie rest for 10 minutes before serving alongside mashed potatoes or steamed greens.