Bring a large pot of heavily salted water to a boil.

Thinly slice 3 large shallots.

Strip the leaves off 1 bunch of collard greens and slice crosswise into 1-inch-wide strips. Discard the stems.

Drop 8 ounces orecchiette into the boiling water, give it a stir, and cook until al dente according to package directions. Scoop out a few cups of pasta water before draining (you'll need quite a bit to help cook down the greens). Drizzle the cooked pasta lightly with olive oil so it doesn't clump together as it sits.

In a large Dutch oven, heat several tablespoons olive oil over medium-high heat. Add the sweet Italian sausage, breaking it into small, half-size chunks as you add it. Cook, undisturbed, until well browned, 3 to 4 minutes. Using tongs, flip the sausage pieces and brown on the second side, 1 to 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 ½ cup tomato paste and 4 tablespoons 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 to 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 you are cooked down and tender (taste one, you'll know), about 8 minutes. Add more pasta water as needed if the pot gets dry. The ragù should be nice and saucy at the end. Remove from the heat and keep covered.

In a large nonstick skillet, heat 3 tablespoons 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 to 4 minutes. Add to the pot of ragù and repeat with the remaining pasta.

Return the pot to medium heat. Add the sausage back in, breaking it into smaller pieces as you stir, with a big splash of the reserved pasta water and a few big handfuls of grated Parm. 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 5 ounces burrata (if using) into small pieces and top each bowl with a few. Drizzle with more olive oil, sprinkle with more Parm, and serve.