Bring 100 ml water seasoning vinegar and sugar and the spices to the boil in a saucepan. Reduce the heat by half. Peel the pearl onions and add them to the still slightly boiling stock and let then steep for 10 minutes. In the meantime, wash the cucumber and cut it with the skin into 1-2 mm thick slices. Remove the pot from the cooker and add the cucumber slices. Put the pot in the fridge. 

Wash the lemon hot, pat dry and finely grate the zest. Place the cream in the saucepan with the Parmigiano Reggiano and heat to 80°C. Remove the pan from the heat and slowly stir in the egg yolks. Add the milk, salt and lemon zest. Leave the Parmigiano Reggiano cream to cool, then whisk and pour into a piping bag. 

Cut the rye rolls in half and toast the cut side over the toaster. Spread a thin layer of the cold butter on the bun halves. Remove the cucumber slices from the pickling stock and top the bun halves with them. Cut off the tip of the piping bag about 5 mm and squirt the Parmesan cream onto the cucumber slices. Halve the pearl onions and press them into the cream, fanning them out. Garnish with the deep-fried potato and parsley chips.