Preheat oven to 450°F (220°C). Spray a large baking sheet with non-stick spray (use the largest sheet you have).
Trim broccoli: remove any tough, woody stem ends. Cut broccoli into pieces about 2 inches long; cut through stems just to where the florets start, then break apart so pieces are roughly the same size.
In a small bowl, whisk together 1 1/2 tablespoons olive oil, 1 tablespoon soy sauce, and 1 teaspoon sesame oil.
Place the broccoli pieces in a large mixing bowl, pour the oil/soy mixture over them, and toss until evenly coated.
Arrange the broccoli in a single layer on the prepared baking sheet, placing any flat sides down so they make good contact with the sheet.
Roast in the preheated oven for 10 minutes.
After 10 minutes, stir or turn the broccoli pieces and continue roasting 5 minutes more, or until the broccoli is tender‑crisp and slightly browned on the edges.
While the broccoli roasts, toast 1 tablespoon sesame seeds in a dry skillet over very high heat for 30–60 seconds, tossing or stirring constantly, until fragrant and lightly browned. Remove from heat immediately.
Transfer the roasted broccoli to a serving dish, sprinkle with the toasted sesame seeds, and serve hot.