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Bring a large pot of heavily salted water to a boil. (Big handful of salt in there, please!) Thinly slice 3 large shallots. Strip the leaves off 1 bunch of collard greens and slice crosswise into 1in-wide strips. Discard the stems.
Drop 8oz of orecchiette into the boiling water, give it a stir, and cook until al dente according to the package directions. Scoop out a few cups of pasta water before draining it (you’ll need quite a bit to help cook down the greens). Drizzle the drained pasta lightly with olive oil so it doesn’t clump together as it sits.
Meanwhile, in a large Dutch oven, heat several tablespoons of olive oil over medium- high heat. Once the oil is very hot, add 1 pound sweet Italian sausage, breaking it into large, golf ball-size chunks as you add it. Cook, undisturbed, until well browned, 3-4 minutes. Using tongs, flip the sausage pieces and brown on the second side, 1-2 minutes. Transfer to a plate.
Reduce the heat to medium-low and add the shallots and a big pinch of salt. Cook, stirring often, until jammy and cooked down, 4 minutes.
Stir in ¼ cup of tomato paste and 4 tablespoons of chopped Calabrian chiles (hold some back if you're spice averse) and stir to evenly coat the aromatics. Continue to cook, stirring, until the tomato paste begins to stick to the pot, 4-5 minutes.
Add the collard greens and a few big splashes of the reserved pasta water (this will help to steam and wilt the greens). Stir well, cover the pot, and cook until the greens are cooked down and tender (taste one, you’ll know!), 4-5 minutes. Add more pasta water as needed if the pot gets dry. The ragu should be nice and saucy at the end. Remove from the heat and keep covered.
In a large nonstick skillet, heat 2 tablespoons of olive oil over medium-high heat. Add about half of the cooked pasta to the skillet and cook, undisturbed, until the bottom is golden brown and crisp at the edges, 3-4 minutes. Add to the pot of ragu and repeat with the remaining pasta. When in doubt, refry.
Return the pot to medium heat. Add the sausage back in, breaking it into smaller pieces as you do, along with a big splash of the reserved pasta water and a few big handfuls of grated parmesan. Cook, stirring vigorously, until a thick sauce coats the pasta, 30 seconds.
Squeeze the juice of 1 lemon into the pasta and stir to combine.
Divide the pasta among serving bowls. Tear 5oz of burrata (if using) into small pieces and top each bowl with a few. Drizzle with more olive oil, sprinkle with more parmesan, and serve.