Preheat oven to 425°F. Drain anchovies and finely chop. Smash, alternating between chopping and mashing with the side of the chef’s knife, until a thick smooth paste forms. Mix with 4 Tbsp. oil in a large bowl. Toss in cauliflower; season very lightly with salt and pepper. Spread out on a rimmed baking sheet. Toss carrots, oregano, and 2 Tbsp. oil on another rimmed baking sheet; season with salt and pepper. Place carrots on upper rack and cauliflower below and roast, tossing halfway through, until carrots are browned and crisp-tender, 14–18 minutes. Remove carrots from oven and grate lemon zest over. Move cauliflower to upper rack and continue to roast, tossing occasionally, until browned all over and tender, 15–20 minutes longer.
Meanwhile, heat 1 Tbsp. oil in a small skillet over medium. Cook panko, stirring, until golden brown. Season lightly with salt and transfer to paper towels to drain.
Just before serving, drizzle carrots with oil and toss cauliflower with breadcrumbs. DO AHEAD: Cauliflower and carrots can be roasted 3 days ahead. Let cool; cover and chill. Bring to room temperature before serving.
