Preheat your oven to 400°F (200°C) and line a large baking sheet with parchment paper.
Slice the chicken breast, carrots, and zucchini into uniform 3-inch sticks to ensure even cooking.
Place the chicken and vegetables on the prepared baking sheet, drizzle with olive oil, and season with half of the salt, pepper, and oregano.
Toss everything together until well-coated, then spread in a single layer across the pan.
Roast for 18-22 minutes, or until the chicken is cooked through and the vegetables are tender and slightly browned.
While the tray is roasting, whisk together the Greek yogurt, lemon juice, minced garlic, and the remaining salt, pepper, and oregano in a small bowl.
Serve the warm roasted sticks immediately alongside the chilled, creamy yogurt dip.