Boil the peeled and chopped potatoes in a large pot of salted water for 15-20 minutes until fork-tender. Drain well, then return to the hot, empty pot for a minute to steam off any excess moisture.

Mash the potatoes with the butter and milk until completely smooth and creamy. Season generously with salt and black pepper, then set aside.

Preheat your oven to 400°F (200°C). In a large skillet or pan over medium-high heat, brown the ground beef until no pink remains, about 5-7 minutes. Drain any excess fat from the pan.

Add the chopped onion, carrots, and celery to the beef. Cook for 5-7 minutes until the vegetables have softened and become fragrant.

Stir in the flour, tomato paste, Worcestershire sauce, and fresh thyme, cooking for 1 minute to cook out the raw flour taste. Pour in the beef broth, bring to a gentle simmer, and cook for 10 minutes until the gravy thickens beautifully.

Transfer the rich beef filling to a 9x9-inch baking dish, spreading it out into an even layer. Carefully spoon the mashed potatoes over the top, smoothing it out or creating decorative peaks with a fork.

Sprinkle the grated cheddar cheese and dried breadcrumbs evenly over the mashed potatoes. Bake in the preheated oven for 20-25 minutes until the breadcrumb topping is golden brown and crispy, and the filling is bubbling around the edges. Let rest for 5 minutes before serving with a side of steamed green peas.