Coat the bottom of a large pot with the oil and place it over medium heat. Give the oil a minute to heat up, then add the onion, carrot, and celery. Cook the vegetables for about 10 minutes, stirring occasionally, until they begin to soften. Stir in the garlic and cook it with the vegetables for about 1 minute, until very fragrant, stirring constantly.
Stir in the broth, cabbage, potatoes, and bay leaf. Raise the heat to high and bring the liquid to a boil, then lower the heat and let the mixture simmer for 25 to 30 minutes, until the vegetables are tender and the broth has reduced a bit.
Stir in the tomatoes and let the soup continue simmering for 5 minutes longer.
Remove the pot from heat. Remove the bay leaf. Stir in the lemon juice and dill, then season the soup with salt and pepper to taste.
Ladle into bowls, sprinkle each with fresh chives, and serve.