Heat the oil in a large skillet over medium-high heat. Add the onions and carrots and cook, stirring occasionally, for 4 minutes or until they start to brown. Add the garlic and cook, stirring, for 30 seconds. Add the lamb and cook until it browns, about 5 minutes. Sprinkle the flour over the lamb and cook, stirring, for 1 minute. Add the tomato paste, chicken broth, steak sauce, Worcestershire sauce, rosemary, thyme, salt, and pepper. Bring to a boil, reduce heat to medium-low and cook, stirring occasionally, for 10 minutes or until the sauce starts to thicken. Stir in the peas and corn.
Put the lamb mixture into a 9-inch square baking dish. Stir the mashed potatoes until smooth and top the lamb mixture with the potatoes, making sure to get the potatoes all the way to the edges to form a seal. Make a cross hatch pattern in the potatoes with the tines of a fork. Top with the Parmesan cheese and sprinkle with paprika. Bake for 25 minutes or until the potatoes start to brown. Let cool for a few minutes before serving.