Toast the wheat berries in a Dutch oven over medium heat, stirring frequently, until they become fragrant and begin to darken, about 5 minutes; transfer the grains to a small bowl.

Heat the oil in the empty pot over medium heat until it shimmers. Stir in the onions and cook until softened and translucent, about 5 minutes. Stir in the garlic and thyme and cook until fragrant.

Add the tomato paste and cayenne, stirring constantly until the paste darkens slightly, about 2 minutes—this develops a deeper savory base for the broth.

Pour in the broth and water, using a wooden spoon to scrape up any browned bits from the bottom of the pot. Stir in the toasted wheat berries, cabbage, beets, carrot, bay leaf, and pepper, then bring the liquid to a boil.

Reduce the heat to low and simmer, partially covered, until the wheat berries are tender but still have a distinct chewy bite and the root vegetables are soft, 45 minutes to 1 hour 15 minutes.

While the soup simmers, whisk the sour cream and dill together in a small bowl and set aside.

Remove the pot from the heat and discard the bay leaf. Stir in the vinegar and salt, then season with additional salt and pepper to taste. Serve hot, topped with a dollop of the dill cream.