Heat 2 tablespoons oil in a large Dutch oven over medium heat. Add chopped onion and sliced garlic; cook, stirring occasionally, until softened and fragrant but not browned, about 4 minutes. Add 4 cups cabbage, 1 teaspoon oregano and ½ teaspoon pepper; cook, stirring occasionally, until the cabbage starts to soften, about 3 minutes.
Add 4 cups broth and the rinsed beans; increase heat to high and bring to a boil. Reduce heat to medium, cover and cook, stirring occasionally, until the cabbage is tender, about 8 minutes.
Meanwhile, whisk ¼ cup Parmesan, 1 egg and 2½ tablespoons lemon juice in a medium heatproof bowl. While whisking constantly, drizzle about ½ cup of the hot soup broth into the egg mixture until completely incorporated. Repeat once more.
Remove the soup from the heat; whisk in the tempered egg mixture. Stir in 2 tablespoons dill and ¼ teaspoon salt. Divide among 4 bowls; garnish with additional dill, if desired.