Preheat the oven to 400 degrees F. Chop the carrots and onions into a medium dice, then grate the garlic cloves and set all of them aside. (To make things easier, I like to cut carrots into quarters like carrot sticks, and then dice those.)
Finely mince the parsley, rosemary, and thyme and set aside. Chop a little extra parsley and reserve to add on top just before serving.
Use a vegetable peeler and a good chef's knife to peel and chop the potatoes into 1" pieces. Place them in a large pot with enough cold water to cover them and 1 tablespoon kosher salt. Place over high heat, once boiling, cook for about 20 minutes until fork tender.
While the potatoes are cooking, add the olive oil to a large, deep skillet over medium heat. Once shimmering, add the diced carrots and onions with a pinch of kosher salt and sauté until the onions are soft and translucent and just beginning to turn brown at the edges, 5-8 minutes.
Add the tomato paste, chopped herbs, and grated garlic to the skillet and stir constantly for 1-2 minutes until fragrant.
Add the lamb to the skillet with 1 teaspoon of kosher salt and some freshly ground black pepper and turn the heat up to medium high. Use a wooden spoon or bamboo spatula to break up the lamb and cook until the fat is rendered and it starts to get crispy, 8-10 minutes.
Make a slurry by mixing the corn starch with a bit of the broth. Add the peas, remaining broth, and slurry to the pan and stir to combine, scraping up any browned bits on the bottom of the pan. Turn the heat to low and simmer for a few minutes until the sauce thickens slightly.
While the filling simmers, finish the mashed potatoes. Drain them in a colander and return to the pot. Add the butter and ½ teaspoon of kosher salt. Gently mash the potatoes until smooth, don't overmix. (I honestly like mine a little lumpy!) Taste for seasoning and adjust if necessary.
Transfer the ground lamb mixture to a 9-in x 9-in baking dish (or other similarly-sized casserole dish). Smooth it out into an even layer. Scoop large spoonfuls of mashed potatoes on top of the lamb filling, then gently spread them out to reach the edges of the dish. Use a fork to make criss cross designs in the top.
Place the dish on top of a sheet pan in case it bubbles over, then bake uncovered for 15-20 minutes, until it's bubbling at the edges and beginning to get golden on top. Let stand for 5 minutes, then top with chopped fresh parsley and serve.