Preheat oven to 350°. Bake 8 oz. prepared cornbread, cut into ¾” pieces, in a single layer on a rimmed baking sheet, tossing once, until golden brown and crisp, 20–25 minutes. Let croutons cool.
Meanwhile, working in 3 batches, cook 2 bunches broccolini (about 1 lb. total), trimmed, and 1 bunch broccoli rabe (about 1 lb.), trimmed, in a large pot of boiling salted water just until bright green and partly tender, about 1 minute per batch. Using tongs, transfer to a bowl of ice water and let cool. Drain and place on a kitchen towel. Pat to remove as much moisture as possible. Cut any thicker stalks into 1″–2″ pieces.
Bring 6 garlic cloves, thinly sliced, and 3 Tbsp. grapeseed or vegetable oil to a bare simmer in a small saucepan over medium-low heat and cook, swirling occasionally, until garlic is just slightly darker in color and very soft, 10–12 minutes. Let cool 10 minutes. Transfer garlic along with oil to a blender. Add 2 large egg yolks, 6 oil-packed anchovy fillets, 3 Tbsp. sherry vinegar or red wine vinegar, 1 Tbsp. tamari soy sauce, 2 tsp. Dijon mustard, and 2 Tbsp. water and purée until smooth. With motor running, stream in 3 Tbsp. grapeseed or vegetable oil; blend until emulsified. Taste dressing and season with salt if needed.
Heat remaining 1 Tbsp. grapeseed or vegetable oil in a large skillet over high. Working in batches, cook broccolini and broccoli rabe in a single layer, tossing occasionally, until tender and blistered in spots, about 3 minutes per batch. Season lightly with salt.
Transfer broccolini and broccoli rabe to a platter. Drizzle generously with dressing (there will be extra), top with croutons, and sprinkle with finely grated dairy-free or regular Pecorino or Gruyère.