Toss the beetroot and potato with a large pinch of salt, tip into a colander in the sink, and leave to drain for 30 mins. Transfer to a clean tea towel and squeeze out any excess liquid. Heat half the oil in a frying pan and fry the onion for 5 mins. Add the grated veg and fry for 10 mins more until just soft. Remove from the heat. Cool slightly. Combine with 1 egg and the flour, season and form into eight patties.
For the herb yogurt, tip everything into a food processor with a pinch of sea salt, and blitz until smooth.
Heat the butter and remaining oil in a large non-stick frying pan over a medium heat, and fry the rosti for 4-5 mins on each side until crisp and cooked through. Put on a baking tray and keep warm in a low oven.
Bring a pan of lightly salted water to a simmer with the vinegar. Stir to create a whirlpool. Crack 1 egg into a cup, wait until the whirlpool has almost subsided, then tip in the egg. Poach for 2-3 mins, remove with a slotted spoon and drain on kitchen paper. Repeat with the other eggs.
Serve the rosti with the herb yogurt, poached eggs, smoked salmon and some extra dill.