Heat olive oil in a large 10- or 12-inch cast iron skillet over medium-high heat. Add ground lamb, season with 1 tsp Kosher salt, then cook, breaking up with a spatula, until evenly browned. Use a slotted spoon to transfer the lamb to a bowl.
Add diced carrots, onion, celery, and thyme sprigs, then season with the remaining Kosher salt and black pepper. Cook until the vegetables have browned slightly and become tender. Stir in tomato paste until dissolved.
Preheat an oven. Bring a large pot of salted water to a rapid boil. Add cubed potato, sweet potato, and parsnip, then cook until tender, depending on how large you cut them. Drain the veggies, then return them to the pot. Mash until very smooth.
Add unsalted butter, of milk, grated white Cheddar cheese, and Kosher salt, then stir until very well mixed. Taste, then adjust seasoning to taste.
Spoon the mashed root veggies on top – this can be a single smooth layer or you can create a decorative top. Transfer to a preheated oven and bake until the top is golden brown and bubbly. Enjoy immediately with fresh cracked pepper and chopped thyme or parsley!