Peel and cube the potatoes, cover with cold salted water and bring to a boil over high heat. Simmer over medium heat until very tender, about 15 minutes. Drain and mash with the milk, butter and salt.
Heat a large deep skillet over medium-high heat. Add the beef, diced onion and carrot and cook, breaking up the meat, until browned and it reaches 71°C (160°F), about 8 minutes.
Stir in the tomato paste for 1 minute, then add the rinsed lentils, broth, Worcestershire sauce and thyme. Bring to a boil, then simmer over medium-low heat, stirring now and then, until the lentils are tender and the mixture is thick, about 25 minutes. Stir in the peas.
Preheat the oven to 200°C (400°F) with a rack in the centre while the filling simmers.
Spread the filling in a 23 x 33 cm (9 x 13 inch) baking dish and top with the mash, spreading it to the edges. Rough up the top with a fork.
Bake for 20 minutes, until bubbling at the edges and golden on top.
Rest for 10 minutes so the filling sets before spooning onto plates.