Place the peeled and quartered potatoes in a large pot and cover with cold water by 1 inch; bring to a boil over high heat, then reduce to a simmer and cook for 15-20 minutes until the potatoes are completely tender when pierced with a fork.
While the potatoes cook, heat a large, oven-safe skillet over medium-high heat; add the ground beef and cook for 5-7 minutes, using a wooden spoon to break it into small crumbles, until no pink remains, then drain off any excess fat.
Add the chopped onion and diced carrots to the skillet with the beef and cook for 5-7 minutes, stirring occasionally, until the vegetables have softened, then stir in the minced garlic and tomato paste and cook for 1 more minute until fragrant.
Pour the beef broth and Worcestershire sauce into the skillet, add the rosemary, salt, and pepper, and bring to a simmer; cook for 5 minutes until the liquid has reduced slightly, then stir in the frozen peas and remove the skillet from the heat.
Drain the cooked potatoes thoroughly and return them to the warm pot; add the warm milk and softened butter and mash with a potato masher until smooth and fluffy, then season with additional salt and pepper to taste.
Preheat your oven to 400°F (200°C); spread the mashed potatoes evenly over the meat filling in the skillet, then use a fork to create decorative ridges on top and sprinkle with the shredded cheddar cheese and paprika.
Bake for 20-25 minutes until the filling is bubbling around the edges and the potato topping is golden brown; let the pie rest for 10 minutes on a wire rack before serving to allow it to set.