Boil the potatoes in salted water until fork-tender, about 15 minutes. Drain well, then mash with 3 tablespoons butter and the warm milk. Season with salt and pepper.
Heat the oven to 400°F. Melt the remaining butter in a large oven-safe skillet over medium heat. Cook the onion, carrots, and celery until softened, 6 to 8 minutes.
Add the lamb and garlic. Cook, breaking up the meat, until no pink remains. Stir in the tomato paste, broth, Worcestershire sauce, thyme, salt, and pepper.
Simmer the filling until slightly thickened, about 8 minutes. Stir in the peas, then spread the mashed potatoes evenly over the top and roughen the surface with a fork.
Bake until the edges are bubbling and the potato peaks are golden, 20 to 25 minutes. Rest for 10 minutes before serving.