In a large Dutch oven, heat 2 tablespoons of olive oil over medium heat. Add 1 chopped onion and 2 thinly sliced garlic cloves, and sauté for about 4 minutes until the onion is translucent and fragrant.

Introduce 4 cups of chopped green cabbage and a sprinkle of oregano into the pot. Cook for another 3 minutes, stirring frequently, until the cabbage starts to soften and wilt.

Pour in 4 cups of vegetable broth and add 1 can of rinsed cannellini beans. Raise the heat to bring the mixture to a gentle boil, then reduce to a simmer. Cover and let it simmer for about 8 minutes until the cabbage is tender.

Whisk together 1/2 cup of grated Parmesan cheese, the juice of 1 lemon, and 1 beaten egg in a separate bowl. Gradually whisk in a ladleful of the hot broth to temper the egg mixture.

Remove the pot from heat and gently stir in the tempered egg mixture along with 1/4 cup of chopped fresh dill. Season with salt and pepper to taste.

Ladle the soup into bowls and garnish with additional dill and a dash of black pepper. Serve immediately.