Preheat oven to 400°F. Position a rack in the center.
Season beef with salt and pepper. Toss with the dredging flour until lightly coated.
Heat oil and butter in a large heavy pot over medium-high. Brown beef in batches; set aside.
Sauté onion, carrots, and mushrooms in the same pot until lightly softened, about 6 minutes.
Stir in garlic and tomato paste. Cook 1 minute, then splash in Worcestershire and scrape up browned bits.
Return beef to the pot. Pour in broth, add thyme and rosemary, and bring to a gentle simmer.
Cover and simmer on low until beef is fork-tender, 70 to 80 minutes, stirring occasionally.
Uncover and simmer 10 to 15 minutes until the gravy thickens. Remove from heat and cool 10 minutes.
Dust the counter with the extra flour. Roll pastry slightly larger than your pie dish.
Spoon the filling into a 9-inch pie dish. Lay pastry over the top, trim, and crimp edges.
Brush crust with milk and cut a few small vents for steam.
Bake 30 to 35 minutes until deeply golden and bubbling. Rest 10 minutes before slicing.
Notes
Variation: Add 1 cup of frozen peas to the filling after thickening for a pop of color and sweetness. For a richer crust sheen, swap the milk brush with a mix of 1 tbsp milk and 1 tsp melted butter.Storage: Cool completely, then cover and refrigerate for up to 3 days. Reheat at 350°F until hot, or freeze the baked pie for up to 1 month.This recipe is an original creation inspired by classic Classic Steak Pie Recipe flavors. All ingredient ratios and instructions are independently developed.