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In a large pot of boiling salted water, cook pasta, stirring occasionally, until very al dente, 6 to 8 minutes (it won’t be fully cooked through). Reserve about 2 cups pasta water.
Meanwhile, in a large skillet over medium heat, heat 1 tablespoon oil. Cook sausage, breaking into large chunks, until golden brown and crispy, 8 to 10 minutes. Using a slotted spoon, transfer to a plate.
In same skillet over medium heat, heat 1 tablespoon oil. Add broccoli; season with 1 teaspoon salt. Cook, mostly undisturbed and reducing heat to medium-low if pan gets too hot, until one side is caramelized and florets are tender with a little crunch, 6 to 8 minutes. Transfer to plate with sausage.
In same skillet over medium-low heat, heat 1 tablespoon oil. Add shallot and garlic and cook, stirring frequently, until shallot is translucent and garlic is fragrant, about 5 minutes. Add cream, tarragon (if using), lemon zest, basil, pasta, and 1 cup pasta water; season with 1 teaspoon salt. Increase heat to medium, bring to a simmer, and cook, tossing and adding more pasta water if needed, until pasta is al dente, 5 to 6 minutes more.
Remove from heat and stir in Pecorino Romano, sausage, and broccoli. Top with burrata. Drizzle with oil; season with pepper.