Set an oven rack about 8 inches from the heat source and preheat the oven's broiler. Toss broccoli in a bowl with 1 tablespoon olive oil and salt; spread on a large baking sheet.
Broil in the preheated oven until broccoli is tender and beginning to brown, 5 to 10 minutes.
Bring a large pot of lightly salted water to a boil. Cook linguine at a boil until tender yet firm to the bite, about 11 minutes.
Heat 1 tablespoon olive oil in a large skillet over low heat. Add anchovies, garlic, and red pepper flakes; cook and stir until anchovies dissolve into the oil, 3 to 5 minutes. Season with pepper. Mix in roasted broccoli.
Drain pasta and return to the pot. Add broccoli-anchovy mixture and mix well. Drizzle remaining olive oil on top.