Preheat the oven to 450 degrees F and line a rimmed baking sheet with parchment paper. Place the drumsticks on one side of the baking sheet, spacing them evenly apart. Place the potatoes on the other side of the baking sheet, leaving an empty space in the center for the broccoli. Drizzle the potatoes with 1 tablespoon of of the olive oil, and sprinkle with salt and pepper, tossing to evenly coat. Bake until the potatoes begin to soften and the drumsticks begin to cook and caramelize, 20 minutes.
Remove the baking sheet from the oven and put the broccoli in the center. Drizzle the broccoli with the remaining 2 tablespoons of olive oil, and lightly sprinkle with salt and pepper, tossing with tongs to evenly coat. Brush the drumsticks with the teriyaki glaze on all sides.
Return the baking sheet to the oven and roast until the broccoli and potatoes are tender and charred in spots and the drumsticks are completely cooked through and read an internal temperature of 160 degrees F, 20 to 25 minutes more. Divide chicken and vegetables between two plates and serve.
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