Put the potatoes in a pan of cold, salted water and bring to the boil. Turn the heat down to a simmer and cook for about 8 minutes, until starting to soften but still quite firm. Drain and leave to cool a little in the colander.
Once the potatoes are cool enough to handle, grate them into a large bowl (leave the skin on, it adds flavour). Pour in the beaten egg and add the flour, herbs and salt. Stir until everything is well combined. Shape the mixture into 4 patties.
Melt the butter in a large non-stick frying pan with the vegetable oil and set over a medium heat. Once hot, add the patties (you may need to cook them in batches) and cook for 3–4 minutes on each side, until golden.
Remove from the pan with a slotted spoon and transfer to plates.
Serve the hash browns topped with scrambled eggs flaked mackerel, and plenty of black pepper.