Prepare the beets.
Peel the beetroots and cut them into thin julienne strips (you should have about 3 cups / 720 ml). Carefully wash the beet stems and tender leaves, then chop them finely (you should have about 2 cups / 480 ml).
Cook the soup base.
In a large, non-reactive pot (like enameled cast iron), combine the julienned beets, chopped stems and leaves, 2 quarts of water, salt, vinegar, and sugar.
Bring to a boil over high heat, then reduce the heat to low, cover, and simmer for about 1 hour, or until the beets are completely tender.
Taste and chill.
Taste the broth and add a little more vinegar for tang or sugar for sweetness if needed.
Remove from the heat and allow it to cool completely. For best results, chill in the refrigerator for at least 4 hours or overnight.
Make the dill cream.
In a medium bowl, combine the chopped fresh dill with a pinch of salt.
Using the back of a spoon or a pestle, gently press and mash the dill until it becomes moist and aromatic.
Add the sour cream and mix well. To prevent curdling, temper the cream by whisking in about 1 cup (240 ml) of the cold beet broth until the mixture is smooth.
Pour this creamy mixture back into the main pot of soup and stir to combine.
Add the finishing touches.
Peel and seed the cucumber, then chop it coarsely. Coarsely chop all 5 hard-boiled eggs.
Stir the chopped cucumber and the contents of 3 to 4 of the chopped eggs into the soup.
Add the beer and serve. Just before serving, stir in the cold beer. Do a final taste test and adjust the seasoning if necessary—a tiny bit more vinegar or sugar can make all the difference.
Serve the soup ice-cold in chilled bowls, adding an ice cube to each for extra chill.
Garnish with the remaining chopped egg.
