Preheat the oven to 190 degrees C.
In a large, deep pan fry the lamb mince until browned. No added oil is necessary, because lamb mince is naturally richer in fat. After initial frying you can put the pan at an angle and remove some fat, if you don't like it.
After this step you need to add all of the finely diced vegetables to the pan and mix well. In the meantime peel and set potatoes for boiling. Cook them until soft and then mash with lactose free milk and season with salt and black pepper.
Season the content of the frying pan with fresh chopped rosemary, salt and pepper. Keep cooking it, covered with a lid and stirring every so often, until the vegetables are soft. Add peas, the Low FODMAP stock and continue cooking on low for a couple of minutes.
Place the meat and veg mix into sprayed roasting dish and cover with mashed potatoes. You can create "hills" with a tablespoon or smooth the potato surface and brush lines with a fork, to create a layer of crispiness. Spray with oil and place in the oven for 30 minutes.
You can prepare the pie in advance, even the day before. Just take it out of the fridge couple of hours before, so it isn't very cold going into the oven and bake in the oven. Time will need to be adjusted and prolonged, because the pie will be completely cooled.