Preheat the oven to 400°F, and line a large sheet pan with a rim with foil.

In a small bowl, stir together the mayonnaise, lemon zest, dried garlic, parsley, and cayenne pepper if using. Add a pinch of salt and pepper to taste. Smear this mixture evenly over both chicken legs. Place the chicken legs on the center of the baking sheet, skin-side up. Roast for 30 minutes.

Meanwhile, cut the carrots into sticks and toss them in a bowl with the broccoli florets. Toss with the olive oil and a pinch of salt and pepper. It might not seem like enough oil at first, but the chicken gives off some oil on the pan while it bakes.

After the chicken has baked for 30 minutes, flip the legs over, then add the vegetables around the edge of the pan. Continue to roast for another 10-15 minutes.

Test the internal temperature of the chicken before eating: it should be at least 165°F in the deepest part of the thigh.