In a large pot over medium heat, brown sausage until crispy, breaking it up using a wooden spoon. Transfer to a bowl and set aside.
Increase heat to medium-high and add 1 tablespoon oil, garlic, onion, and thyme; season with salt and pepper. Sauté until softened, about 4 minutes.
Add chickpeas, white wine, cream, and stock; bring to a boil and simmer until chickpeas are tender, 5 to 8 minutes.
Toss in broccoli and sausage; simmer until broccoli is bright green and cooked al dente, about 2 minutes. Garnish with parsley and drizzle with olive oil. Serve immediately.