Fill a large pot with water. Bring to a boil. Once boiling, salt the water. Working in batches if necessary, cook the kale until just tender, about 4 minutes. Using tongs, transfer cooked kale to a baking sheet and let cool. Once cool, squeeze out any excess liquid from the kale. Finely chop leaves and stems, then set aside.
In a medium-sized pan, heat 3 tablespoons olive oil over medium heat. Add breadcrumbs and cook, stirring often, until they start to brown. Add one-third of the garlic and cook until breadcrumbs are golden. Transfer to a paper-towel-lined plate and let cool. Bring the large pot you cooked the kale to a boil and make sure it is seasoned with salt. Add the orecchiette and cook until al dente.
Meanwhile, heat butter and remaining olive oil in the medium-sized pan. Add remaining garlic and chili flakes. Cook for a minute or two. Add kale and 1/2 C water. Cook, stirring often until kale is warmed through, about 4 minutes. Season with salt and pepper. Once the pasta is al dente, reserve 1 cup of the cooking liquid and add drained pasta to the kale mixture. Add 1/2 C pasta water and stir to coat, add more pasta water as necessary, until the sauce coats the pasta.
Mix in the Parmigiano Reggiano and 1/2 of the breadcrumbs. Toss to combine and quickly divide the pasta between the bowls. Drizzle with a little extra virgin olive oil and finish with the remaining breadcrumbs.