Heat 1/4 cup olive oil in a large, heavy-bottomed Dutch oven over medium heat. Add the diced onions, carrots, and celery, stirring occasionally until they begin to soften, about 8-10 minutes. Allow the vegetables to caramelize slightly at the edges to develop natural sweetness.
Add the minced garlic, dried thyme, rosemary, red pepper flakes (if using), and bay leaf to the softened vegetables. Cook while stirring constantly for about 30-45 seconds, until fragrant but not browned.
Pour in the diced tomatoes with their juices and stir to combine, scraping up any flavorful bits from the bottom of the pot. Add the vegetable broth and bring the mixture to a gentle boil. Reduce heat to maintain a simmer, then cover partially and cook for about 15 minutes.
Add the drained cannellini beans, chopped kale, and cabbage to the pot. Stir to incorporate all ingredients, then simmer for an additional 20-25 minutes until the greens are tender but still retain some structure.
Add the torn bread chunks to the soup, pressing them down to ensure they're submerged in the broth. Cover and remove from heat, allowing the bread to soften and absorb the flavorful liquid for about 10 minutes.
Gently stir the soup to incorporate the softened bread throughout. If it seems too thick, add a splash more broth or water. Stir in the grated Parmesan cheese, then taste and adjust seasoning with salt and pepper.
Ladle into warmed bowls and finish each serving with a generous drizzle of your best olive oil and additional Parmesan if desired.
For the most authentic experience and best flavor, cool completely after cooking, refrigerate overnight, then gently reheat (or "reboil," as the name suggests) before serving.