Preheat the oven to 375° F. Bring a large pot of heavily salted water to a boil over high heat.
Toss the breadcrumbs with 1 tablespoon olive oil. Spread them on a baking sheet and toast 8 to 10 minutes, stirring once or twice, until golden brown and crispy.
Spread the pine nuts on another baking sheet, and toast them 4 to 5 minutes, until they're golden brown and smell nutty. Crush half the pine nuts, and combine them with the whole pine nuts, breadcrumbs, and parsley in a small bowl. Season with salt and pepper.
Blanch the broccoli in the rapidly boiling water 2 to 3 minutes, until just tender. Drain, and cool on a baking sheet.
Heat the remaining olive oil and the butter in a small saucepan over low heat. Add the anchovies and chili and cook 5 minutes, stirring with a wooden spoon as the anchovy melts into the sauce. Add the garlic and thyme and turn off the heat. Season with 1/2 teaspoon salt.
Cut the burrata into 6 slices and arrange on 6 dinner plates.
Heat a large sautè pan over high heat for 1 minute. Add the anchovy butter, shallots, and broccoli and season with salt, pepper, and a squeeze of lemon. Toss well to warm the broccoli and coat it with the anchovy butter.
Taste for seasoning and plate next to the burrata. Shower the pine nut breadcrumbs over the top.