Bring a large pot of heavily salted water to a boil.
Meanwhile, cut the broccoli into bite-size florets. Trim off the tough parts of the stem and discard. Thinly slice the stem crosswise. Peel and thinly slice the garlic.
Add the pasta to the boiling water. About 2 minutes before the pasta is al dente, add the broccoli florets and cook until the pasta is ready. Turn off the heat and scoop out 1 cup of pasta water, then drain the pasta.
Transfer the pasta and broccoli florets to a baking sheet, drizzle with oil, and toss to combine. Spread the mixture into an even layer (this will cool down the pasta a bit and prevent it from sticking).
In a food processor fitted with the blade attachment, add the cottage cheese, Parmesan, lemon zest and juice, 1/2 teaspoon kosher salt, and several grinds black pepper. Process, scraping down the bottom and sides of the bowl as needed, until smooth. With the motor running, stream in a splash of the reserved pasta water until the sauce reaches your desired consistency.
Heat 2 tablespoons olive oil in the now-empty pasta pot over medium heat. Add the garlic and broccoli stems and sauté until the stems are tender but still have a slight bite, 4 to 5 minutes. Remove from the heat and stir in the pasta and prepared sauce. Stir, adding another splash of pasta water if needed, until the pasta is creamy and well-coated. Taste and season with salt and pepper.
Serve with more Parmesan.
