Brown the ground lamb. Heat 2 Tbsp olive oil in a large 10- or 12-inch cast iron skillet over medium-high heat. Add 1½ lbs ground lamb, season with 1 tsp Kosher salt, then cook, breaking up with a spatula, until evenly browned, about 4-6 minutes. Use a slotted spoon to transfer the lamb to a bowl.
Sauté the aromatics. Add diced carrots, onion, celery, and 4 fresh thyme sprigs, then season with the remaining ½ tsp Kosher salt and ¼ tsp black pepper. Cook, stirring occasionally, for about 6-8 minutes, or until the vegetables have browned slightly and become tender. Stir in 2 Tbsp tomato paste until dissolved.
Create the sauce. Deglaze the pan with ½ cup (118ml) red wine, scraping the bottom of the pan as needed. Allow this to reduce by about half, then return the ground lamb to the pan and add 2 Tbsp Worcestershire sauce, 1 Tbsp Dijon mustard, 1½ cups (355ml) low-sodium beef broth, and 2 bay leaves.
Simmer. Reduce the heat to medium-low, then simmer for about 20 minutes. Discard the thyme sprigs and bay leaves.
Cook the root vegetables. Preheat an oven to 400F. Bring a large pot of salted water to a rapid boil. Add cubed potato, sweet potato, and parsnip, then cook until tender, about 10-15 minutes, depending on how large you cut them. Drain the veggies, then return them to the pot. Mash until very smooth.
Add cheese. Add 4 Tbsp unsalted butter, ⅓ cup (79ml) of milk, 1 cup (4oz) of grated white Cheddar cheese, and 1 tsp Kosher salt, then stir until very well mixed. Taste, then adjust seasoning to taste.
Adjust seasoning of the filling, if needed. When ready to assemble, give the filling another taste to make sure it’s seasoned well. This is your chance to change anything you don’t like. Stir in 2 cups of frozen peas.
Bake until golden brown. 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, about 35-40 minutes. Enjoy immediately with fresh cracked pepper and chopped thyme or parsley!