Bring the vegetable stock to boil in a saucepan. Stir in the barley. Lower the heat, cover, and simmer on very low heat until the barley is tender and the liquid is absorbed. Let stand about 5 minutes before adding the leeks and walnuts.
While the barley cooks, heat the first 2 tablespoons of oil in a skillet over medium flame until shimmering. Add the leeks and sauté until tender, about 5 minutes. Season with salt and a generous grind of black pepper.
Gently stir the leeks and walnuts into the cooked barley. Keep warm while you prepare the broccoli.
Cut the broccoli florets from the thick stalk, leaving about an inch of stalk attached to the florets. Save the rest of the stalks for another purpose, perhaps a stir-fry. Cut the florets through the base of the head with a small knife, pulling the florets apart. Cut the larger florets in half down the middle.
Heat the remaining olive oil in the skillet. Add the broccoli florets and sauté a minute or so. Add a few tablespoons of water, cover, and cook several minutes, until the broccoli is just tender. Season with salt and pepper to taste. Serve on top of the pilaf.