Bring a large pot of salted water to a boil, then add the pasta and cook until just shy of al dente, about 8 minutes. Drain, reserving about half a cup of pasta water.
While the pasta cooks, chop the broccoli into small florets. In a skillet over medium heat, heat a tablespoon of olive oil and add the broccoli. Sauté until tender and slightly charred, about 7-8 minutes, filling your kitchen with a smoky aroma.
Add the minced garlic to the skillet with the broccoli, and sauté for about 30 seconds until fragrant, being careful not to let it brown too much. The garlic should smell fragrant and golden.
Transfer the cooked broccoli and garlic to a blender. Pour in the reserved pasta water, add the milk or sour cream, and squeeze in the lemon juice. Blend until smooth and creamy, tasting and seasoning with salt and pepper as needed.
Pour the blended sauce back into the skillet over low heat. Warm gently, stirring often. If the sauce feels too thick, add a tablespoon of pasta water at a time until it reaches a silky, velvety consistency that coats the back of a spoon.
Add the drained pasta to the skillet and toss to coat evenly in the sauce. Cook together for 1-2 minutes to allow flavors to meld and the pasta to absorb the sauce. The pasta should be tender, shiny, and coated in a luscious, greenish sauce.
Remove from heat, then stir in freshly grated Parmesan cheese if using. Finish with a drizzle of olive oil and a squeeze of lemon for brightness. Taste and adjust seasoning with salt and pepper.
Serve immediately, with extra lemon wedges on the side if you like a zesty finish. The dish should be vibrant, velvety, and satisfying, with a hint of brightness from the lemon.
