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.
Photograph by Ryan Dausch