Heat the oil in a large pot over medium-high heat until shimmering. Stir in the onion, carrots, celery, garlic and a pinch of salt. Stir to coat. Cook, stirring occasionally, until tender, about 10 minutes.
Add the kale, a big handful at a time, stirring to wilt before adding more. Stir in the broth, tomatoes and beans. Add parmesan cheese rinds, if available. Simmer until the kale is tender and the beans have lost their canned taste, about 15 minutes. Stir in the vinegar or lemon juice. If desired, season with black pepper and red pepper flakes. Keep warm over very low heat.
Arrange a rack in the middle of the oven and heat the broiler. Cut the loaf of bread in half lengthwise. Brush the cuts sides of bread lightly with olive oil and sprinkle on a dash of salt and pepper. Place bread, cut side down, on a baking sheet. Broil until lightly browned, about 3 minutes. Flip the bread over so that the cut sides now face up. Cover cut sides with shredded Parmesan cheese. Broil, watching carefully, until the edges of the bread are crisp and the cheese is browned and bubbly, about 5 minutes more Cool slightly and slice . Place 1-2 slices of bread in each serving bowl and ladle in the stew. Serve hot.