Cut the chicken breasts into 2–3 cm pieces and combine them in a large bowl with the olive oil, lemon juice, garlic, oregano, salt, and pepper; cover and refrigerate for 30–60 minutes. The chicken should look evenly coated and glossy, with oregano clinging to each piece.

Blend the tzatziki olive oil with the minced garlic, then grate the peeled and seeded cucumber into a bowl; season it and let it sit for 10 minutes before squeezing it dry in a towel. Stir the cucumber with the garlic oil, strained yogurt, red wine vinegar, and salt until thick, then refrigerate the sauce until serving.

Trim the wooden skewers to fit the cooking surface if needed, soak them in water, and thread on the chicken without packing the pieces tightly. Leave small gaps between the cubes so the surfaces can brown instead of steaming.

Heat a grill or griddle pan over high heat and cook the skewers for about 8–10 minutes, turning them regularly. Look for golden-brown char marks on every side and firm but yielding chicken before removing the skewers from the heat.

Heat the oven to 250°C, brush the pita breads with olive oil, season with salt and oregano, and warm them on an upside-down baking tray for 2 minutes. The pita should feel warm and flexible enough to fold without cracking.

Arrange the cooked skewers with sliced red onion, a drizzle of extra virgin olive oil, and lemon juice, then serve with warm pita and chilled tzatziki. For wraps, spread tzatziki over pita and add chicken, tomato, onion, and fried potatoes if using before folding snugly.