Heat olive oil and butter in a large Dutch oven or deep skillet over medium-high heat until melted and shimmering, about 2 minutes.

Pat dry the beef chunks with paper towels. Season with salt and pepper, then sear the beef in batches until browned on all sides, about 3-4 minutes per batch. Remove and set aside.

Add diced onions to the pan, stirring until they soften, about 4-5 minutes. Add minced garlic and sliced mushrooms, cooking until the mushrooms release their moisture and turn golden brown, about 5-7 minutes.

Stir in the tomato paste and cook for 1-2 minutes.

Sprinkle flour over the mixture and stir for 1 minute.

Slowly pour in beef broth while whisking to combine. Return the seared beef and add thyme. Simmer covered for 30-40 minutes.

Incorporate diced carrots and cook for an additional 10 minutes, then stir in the green peas during the last few minutes.

Preheat your oven to 400°F (200°C). Transfer the filling to a deep baking dish.

Roll out the puff pastry to fit over your dish. Lay it atop the filling and cut slits in the center for steam to escape. Brush with egg wash.

Bake for 25-30 minutes or until the crust is golden brown and puffed up. Let rest for 5 minutes before serving.