Bring a large pot of salted water to a boil. Add the pasta and cook as the label directs, adding the grated broccoli during the last 30 seconds of cooking. Reserve 1 cup cooking water, then drain.
Meanwhile, combine the breadcrumbs and 2 teaspoons olive oil in a small nonstick skillet over medium heat. Season with salt and cook, stirring occasionally, until golden brown, about 5 minutes; remove to a bowl.
Heat the remaining 3 tablespoons olive oil with the garlic in a separate pot over medium-high heat. Cook until the garlic is just golden, about 3 minutes. Add the red pepper flakes, clams and wine. Cover and cook until the clams open, 5 to 7 minutes; remove the clams to a bowl using a slotted spoon (discard any unopened clams). Add the butter, pasta, broccoli and 1/2 cup of the reserved cooking water to the pot; cook, stirring, until the pasta is well coated, about 1 minute. Gradually add the remaining 1/2 cup cooking water if the pasta seems dry.
Stir the clams and any juices into the pasta. Drizzle each serving with olive oil and sprinkle with the breadcrumbs.