Finely chop the black pudding and cook in a dry nonstick frying pan over medium heat. Mash it into crumbs with a fork as it softens and then fry until it just starts to crisp (1 to 2 minutes). Set aside and keep warm.
Whisk the eggs with milk, sea salt and pepper.
Heat the butter in a nonstick pan, add the eggs, and cook over medium heat. Don't stir, just slowly push the eggs around the pan in a broad sweep with a wooden paddle until just cooked, soft and curdy (1 to 2 minutes).
Pile the eggs onto hot buttered toast. Toss the black pudding with dill and scatter on top.