Prep the skewers. If using wooden skewers, soak them in water for at least 20–30 minutes so they don’t scorch on the grill or in the oven.
Marinate the chicken. In a large bowl, whisk the Greek yogurt, Parmesan, olive oil, garlic, lemon juice, lemon zest, Italian seasoning, smoked paprika, salt, and pepper.
Add the chicken cubes and toss until well coated. Cover and refrigerate for at least 30 minutes (up to 4 hours for deeper flavor).
Heat your cooking method. For grilling, preheat to medium-high heat (about 400–425°F). For oven roasting, preheat to 425°F and line a sheet pan with foil or parchment.
Set a wire rack on the pan if you have one for better air circulation.
Assemble the kabobs. Thread chicken onto skewers, alternating with onion and bell pepper if using. Avoid packing too tightly so heat can circulate and brown the edges.
Cook on the grill. Place skewers on the grates. Cook 10–12 minutes total, turning every 3–4 minutes, until the chicken is cooked through and lightly charred in spots.
Internal temperature should reach 165°F.
Or roast in the oven. Arrange skewers on the prepared pan. Roast 12–16 minutes, turning once halfway through, until the chicken hits 165°F and edges are golden.
Finish and serve. Sprinkle with a little extra Parmesan and chopped parsley. Squeeze fresh lemon over the top.
Taste and add a pinch of salt if needed. Serve hot.