In a large bowl, combine the grated potatoes and onions with the beaten egg, flour, and 1 teaspoon of salt.
In a skillet, heat about 1 tablespoon oil over medium heat until shimmering. Form the mixture into pancakes and fry in batches until brown and crispy golden on both sides. You may need to add more oil to the pan between batches.
Transfer to a paper towel-lined plate, season with salt and pepper. Keep the latkes warm by placing them on a sheet pan in an oven set to low.
Prepare the Hollandaise by melting the butter over a double boiler in a heatproof bowl set over a pan of simmering water. Spoon off the white solids, then pour the reserved clear yellow liquid into a separate container. You can also melt the butter in the microwave and use a fat separator to remove the solids.
In another bowl, whisk the egg yolk and water, then place it over the double boiler, whisking constantly.
When the eggs begin to froth, gradually add the clear liquid butter in a slow and steady stream, continuing to whisk. After the sauce has the consistency of thin mayonnaise, remove it from the heat, then add lemon juice, salt, and dill to taste.
To warm the Hollandaise just before serving, heat it very gently over the double boiler, whisking constantly. Be careful not to overheat or the Hollandaise may lose its creamy consistency.
FOR POACHED EGGS:
Using the same pot of simmering water from the double boiler, add vinegar and keep the water at a simmer (do not boil).
Break the eggs into individual ramekins or small cups.
Gently stir the simmering water in a circular fashion, then slide the eggs into the water one at a time. You will need to add them in batches.
Cook until the eggs whites are set. Remove with a slotted spoon and pat dry on a paper towel. Season with salt and pepper.
Plate the dish by placing lox atop 1 to 2 latkes. Add a poached egg followed by warm Hollandaise sauce.