Preheat oven to 425 degrees F. Line a baking sheet with parchment or foil and lightly grease.
Toss potato chunks with olive oil, garlic powder, paprika, salt, and pepper. Spread cut-side down on one half of the baking sheet.
Pound chicken to even thickness. Season both sides with salt and pepper. Mix yogurt with garlic powder, onion powder, and Dijon if using. Coat chicken thoroughly.
In a shallow bowl, combine breadcrumbs, shredded cheese, Italian seasoning, paprika, cayenne if using, and oil or butter. Press yogurt-coated chicken into the mixture to coat both sides. Place on the other half of the baking sheet.
Roast 25 to 30 minutes, flipping potatoes halfway through, until chicken reaches 165 degrees F and breading is golden and set. Broil 1 to 2 minutes at the end if desired.
While baking, whisk together all herb yogurt sauce ingredients. Refrigerate until ready to serve.
Rest chicken 5 minutes before serving. Plate with roasted potatoes and sauce on the side.