Heat your oven to 350°F and grab a large skillet. Brown the 2 lbs of lamb mince over medium-high heat for about 5-6 minutes, breaking it apart as it cooks. You'll hear it sizzle and get a nice golden color—that's exactly what you want. Drain off any excess fat once it's cooked through, then set the meat aside on a plate.
In the same skillet, add the diced onion and cook for 2-3 minutes until it starts to soften. Add the minced garlic and cook for another 30 seconds until fragrant. Stir in the diced carrots and let everything cook together for about 3 minutes.
Return the lamb to the skillet and stir in the tomato paste, cumin, and smoked paprika. Mix everything together so the spices coat the meat and veggies evenly—this usually takes about 1-2 minutes. You'll smell how amazing it is at this point.
Pour in the beef broth and add the peas and corn to the mixture. Bring it to a simmer and let it bubble away for about 8-10 minutes until the liquid reduces by half. This shepherd's pie recipe gets its depth of flavor from this simmering time, so don't skip it.
While the filling simmers, peel and cube the 2 cups of potatoes, then boil them in salted water for about 12-15 minutes until they're fork-tender. Don't panic if they look a little watery at first after draining—that's totally normal.
Mash the hot potatoes with the butter, milk, and salt until you get that creamy texture (I like mine with just a few small lumps). Taste it and adjust the seasoning since you want it slightly salty to balance the filling underneath.
Pour the meat mixture into a 9x13-inch baking dish and spread the mashed potato topping over it evenly. Bake for about 25-30 minutes until the top turns golden brown and you can see the filling bubbling around the edges. Let it sit for 5 minutes before serving so everything sets up nicely.