Prep the potatoes first. Heat the oven to 200C/390F/gas 6. Put a potato on to a tablespoon or wooden spoon and, using a small, sharp knife, make incisions at 2-3mm intervals along the potato without cutting all the way through – the spoon helps stop your knife cutting all the way down. Repeat with the rest of the potatoes.
Melt the butter in a small pan. Put the potatoes in a baking tin, season with salt and pepper, and toss in the butter, thyme and oil. Roast for 35-45 minutes, (the time will depend on the size of the potatoes), basting the potatoes in the fat a few times during cooking.
Meanwhile put the anchovies, capers, garlic, egg yolks and oil in a medium-sized bowl and blitz to a smooth emulsion with a handheld blender. Taste and check for seasoning – the anchovies will provide the salt, but a few generous grinds of black pepper will make all the difference. Line a baking tray with greased baking paper or foil, and put the fish fillets on top.
Wash the parsley, pick the leaves (discarding the stems) and finely chop. Toss into the breadcrumbs and chilli flakes. Spread a heaped teaspoon of the anchovy emulsion over the skin of each fillet, followed by a generous layer of parsley breadcrumbs. When the potatoes are almost ready, drizzle the fillets with a little oil (about two teaspoons), then roast alongside the potatoes as the potatoes for 10-12 minutes (or longer, if you have a thicker fillet of cod or hake), until the breadcrumbs are golden and toasted and the fish gently flakes. Serve the fish and the golden potatoes with a crisp, green salad and a wedge of lemon.