Step 1

Whisk garlic, vinegar, peppers, mayonnaise, sugar, dried oregano, and ½ cup oil in a medium bowl; season with salt and pepper. Transfer half of dressing to a small bowl; cover and chill until ready to serve.

Step 2

Cut chicken thighs lengthwise into long strips, about 1"–1½" wide (you should get 2–3 pieces per thigh). Combine chicken and remaining dressing in a large resealable plastic bag. Seal bag, pressing out any air. Turn to coat and chill at least 20 minutes and up to 1 day.

Step 3

Prepare a grill for medium-high heat; oil grate. Remove chicken from dressing; discard dressing. Thread 2 pieces of chicken onto each skewer. Season chicken lightly with salt and grill, turning occasionally, until golden brown and cooked through, 8–10 minutes.

Step 4

Meanwhile, drizzle onion with oil; season with salt. Grill until browned on both sides but still firm in center, about 3 minutes per side. Transfer to a large bowl and add tomatoes and a couple splashes of reserved dressing. Toss to coat; season salad with salt and pepper. Top salad and chicken with oregano leaves.

Step 5

Serve chicken with salad, reserved dressing, and lemon wedges for squeezing over.