Pour the olive oil in a Dutch oven over medium heat and allow it to heat up just a bit. Add the butternut squash, carrots, celery and onions and sprinkle lightly with salt and pepper. Saute until they develop some color and cook down a bit, about 10 minutes. Add the garlic and cook for 1 minute. Add the beans, tomatoes, Parmigiano rind and 3 cups of the stock. Bring to a boil, reduce to a simmer and cook for 15 minutes. Remove the Parmigiano rind. Add the cabbage and kale and simmer for an additional 20 minutes. Check the soup for seasoning and adjust to taste. Add the bread and press it into the broth to absorb. Simmer for 5 more minutes. If the bread still seems dry, add the remaining cup stock (it should have the look of bread pudding).
Cool the ribollita to room temperature before placing it in a large bowl. Cover and refrigerate for a few hours.
Preheat the oven to 200 degrees F. Add 2 tablespoons light oil to the bottom of a 10-inch nonstick skillet and preheat it over medium-high heat until hot. Using two spoons, spoon 1 1/2 to 2 cups ribollita into the skillet to cover the bottom. Flatten it a bit and let it cook until both sides are deeply golden brown and crispy, 4 to 5 minutes on each side. (Don't worry if the ribollita breaks as you are flipping it with your spatula; just continue to cook until browned on both sides.) Transfer to a baking sheet and keep warm in the oven while you cook the remaining ribollita in the same manner.
For serving: Add the crispy ribollita to a platter. Top with the burrata, freshly grated Parmigiano-Reggiano and freshly ground black pepper. Cut into wedges and serve the prosciutto alongside.