Preheat oven to 400°F (200°C). Toss the fingerling potatoes with olive oil, oregano, salt, and pepper. Spread them on a baking sheet and bake for 20 minutes.
While the potatoes bake, prepare the marinade for the chicken. In a bowl, combine yogurt, smoked paprika, garlic, and a pinch of salt and pepper. Add the chicken breasts and toss to coat them evenly.
After 20 minutes, remove the potatoes from the oven and add the bell peppers to the baking sheet. Push the potatoes to one side and arrange the marinated chicken breasts on the other side, with the sliced red onion scattered around.
Return the baking sheet to the oven and bake for 20-25 minutes, or until the chicken is cooked through and the vegetables are tender.
While the chicken bakes, prepare the tzatziki. In a small bowl, combine the grated cucumber with the yogurt, lemon juice, and a pinch of salt and pepper. Stir well and set aside.
Once the chicken is cooked through, remove the baking sheet from the oven and let it rest for a few minutes.
Serve the chicken breasts with the roasted vegetables and top with a generous dollop of homemade tzatziki. Drizzle with any leftover pan juices, if desired.