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Heat the oil in a large saucepan or small pot over medium-high heat until shimmering. Add the onion, garlic, and salt, and stir to coat. Cook, stirring often, until tender and golden-brown, about 15 minutes.
Stir in the thyme, sage, parsley, peppercorns, and pickling spice, and cook 1 minute while stirring. Add the wine and scrape up any browned bits from the bottom of the pot. Simmer until reduced by half, about 5 minutes.
Stir in the broth or water and Parmesan rinds. Bring to a boil. Reduce the heat to low, then simmer until the liquid reduces by half, about 2 hours.
Strain through a fine sieve into a large bowl, pressing on the solids. Discard the solids (although some people cannot resist scraping the softened cheese from the rinds for a cook's treat). You should have about 4 cups (if you are short, add water to get to 4 cups). Use within 1 hour, or cool to room temperature, cover, and refrigerate for up to 3 days or freeze in an airtight container for up to 3 months. If refrigerated or frozen, let come to room temperature before using.
Arrange a rack in the middle of the oven and heat the broiler. Heat the oil in a large pot or Dutch oven over medium-high heat until shimmering. Stir in the onion, carrots, celery, garlic, and a big pinch of salt, and 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 reserved Parmesan broth, tomatoes, beans, and another pinch of salt. Simmer until the kale is tender and the beans have lost their canned taste, about 15 minutes. Stir in the vinegar and season generously with salt, pepper, and pepper flakes. Keep warm over very low heat.
Brush both sides of the bread with olive oil and sprinkle with salt and pepper. Arrange in a single layer on a baking sheet. Broil until browned on top, about 3 minutes. Flip the bread and cover the other side of the bread with Parmesan curls. Broil until the edges of the bread are crisp and the cheese is browned and bubbly, about 5 minutes more. Place a slice in each serving bowl and ladle in the stew. Serve hot.