Preheat the oven to 200C/400F/Gas 6. For the champ topping, heat a pan of salted water, add the potatoes and bring gently to the boil, then simmer until the potatoes are nearly cooked through, about 15-20 minutes.
Meanwhile, for the beef mixture, heat half the olive oil in a large frying pan over a medium heat and gently fry the chopped onion, garlic, carrots and celery for 5-10 minutes, or until softened.
Add the remaining beef mixture ingredients except for the mince and season to taste with salt and freshly ground black pepper. Bring to the boil, then reduce the heat slightly and simmer for 4-5 minutes until the vegetables are tender. Keep warm.
Heat the remaining olive oil in a separate frying pan over a medium heat, add the mince in batches, if necessary, and fry until golden-brown. Stir the tomato sauce into the beef.
When the potatoes for the champ topping are almost done, add the frozen peas and cook for a few more minutes until the peas are tender.
Melt the butter in a small saucepan over a low heat, then gently fry the spring onions until softened. Add the milk and heat through.
Drain the potatoes and peas and mash roughly. Add the warm milk mixture to the potatoes and continue to mash until combined but still chunky.
Spoon the beef mixture into a medium pie dish and top with the champ. Scatter over the cheese and smoked paprika.
Place the dish onto a baking sheet and bake the pie for 25-30 minutes, or until the potato is golden-brown.