Heat the oil in a frying pan, and add the grated carrot and diced onion. Cook over a medium heat for around 5 minutes, until fairly soft. Add the minced garlic, smoked paprika and dried herbs, and cook for 2 more minutes.
Next, add the tomato sauce, tomato puree and about two thirds of the cherry tomatoes (save some back for topping), along with a good pinch of salt and pepper. Bring to a gentle simmer, and cook for a few minutes.
Thinly slice the potatoes - I didn't bother peeling mine. If you're not planning to pre-boil the potatoes, make sure you don't cut them any thicker than a few millimetres. Or, if you want to cut them a bit thicker or you're pushed for time, pre-boil them until they're just tender.
Add one spoonful of sauce to the bottom of a baking dish (mine measured 8 x 8 inches), and spread it around - this just stops the potatoes from sticking to the bottom of the dish. If you have a dish with a lid, that's ideal.
Now layer the sliced potatoes, tomato sauce and grated cheese - I ended up with 4 layers of each in total. Just make sure the potatoes are always laid out in a single layer, to help them cook through (less important if you've pre-boiled them).
When you've used all the potatoes, finish with one final layer of tomato sauce, the last bit of grated cheese, and the breadcrumbs, as well as any cherry tomatoes you held back earlier.
If you didn't pre-boil the potatoes, add a lid to the dish (or cover with kitchen foil, if you don't have a lid), and bake at 190°C (Gas Mark 5 / 375°F) until the potatoes are soft (check by poking with a sharp knife). Mine took around an hour and a half in total, but the exact time will depend on the exact thickness of your potatoes.
If you did pre-boil the potatoes, you can cook your potato bake at 190°C (Gas Mark 5 / 375°F) without a lid, until golden brown and crispy - around 20-25 minutes.