My wife would often complain about the stems when I would prepare kale for us. Not wanting to waste them, I looked for a way to incorporate them that would be to her liking. The goal here was to make the kale stems the highlight of the dish. —Derek
Wash leaves thoroughly under cold running water. Shake off excess water. Using a sharp knife, separate the entire stem from each leaf, reserving both the leaves and stems in separate piles.
Trim the bottoms of the stems if necessary. Lay stems flat in a single bunch and cut through the center to separate the small upper portion from the fat bottom portion. Take the half with the large bottom stems, and carefully split each stem piece in half lengthwise with a sharp knife. Lay the stem pieces down horizontally and slice them very thin (about 2 mm), working with no more than 5-6 pieces at a time.
Heat the butter in a large (about 12") skillet that can be covered later with a lid. Meanwhile, mince the shallot. When the butter has just started to brown, add the shallot and all of the stem pieces. Immediately turn the heat to low and stir gently to evenly distribute across the bottom of the pan.
Adjusting the heat as necessary, saute the shallot and stem pieces very slowly until caramelized, crispy, and very aromatic, about 45 minutes. Stir no more than once every 15 minutes to promote caramelization. Monitor the heat carefully to prevent burning. Towards the end of the cooking process, chop the reserved kale leaves crosswise into 1 inch strips.
Remove stems and any remaining liquid to a separate bowl. Increase heat to medium high and bring the white wine to a boil. Allow to boil for 1-2 minutes to burn off the alcohol, then add the strips of kale leaves, a generous pinch of coarse kosher salt, a small pinch of red pepper flakes, and a turn of fresh ground black pepper. Stir quickly to combine, then cover the skillet and reduce heat to low. Cook covered for 2-3 minutes, shaking the pan once or twice during cooking.
Remove the lid from the skillet, and transfer the kale leaves to a serving bowl using tongs. Dress with a drizzle of walnut oil, white wine vinegar, and a few drops of fresh lemon juice. Toss gently, then cover with the caramelized stems from the bowl, making sure to scrape in any liquid from the bowl. Cover with freshly grated Parmigiano-Reggiano cheese. Serve immediately.