In a bowl, whisk together 1/4 cup olive oil, 3 tablespoons lemon juice, 3 pressed garlic cloves, 1 teaspoon ground cumin, 1 teaspoon paprika, 1/2 teaspoon salt, 1/2 teaspoon black pepper, and 1/4 teaspoon turmeric to make the marinade.
Place the 4 chicken thighs in a resealable plastic bag or shallow dish and pour the marinade over them. Seal or cover and marinate in the refrigerator for at least 30 minutes and up to 2 hours.
While the chicken marinates, make the yogurt sauce: combine 5–6 ounces plain Greek yogurt, 1 teaspoon lemon juice, 1 pressed garlic clove, 1/4 teaspoon salt, and black pepper to taste in a small bowl; stir until smooth and refrigerate until serving.
Preheat the oven to the warm setting (or the lowest setting) and place the pita inside to warm later.
Heat a large skillet over medium-high heat. Remove the chicken from the marinade and discard any excess marinade. Add the thighs to the hot skillet and sear 2–4 minutes per side until golden and an instant-read thermometer registers 165°F in the thickest part. Reduce heat if the chicken is browning too quickly.
Transfer the cooked thighs to a plate and tent with foil to keep warm.
With the skillet over medium-high heat, pour in about 1/4 cup chicken broth or water to deglaze, scraping up browned bits from the bottom as the liquid simmers.
Add the red onion wedges to the skillet and cook about 3–5 minutes, stirring occasionally, until they reach your desired tenderness.
Slice the rested chicken thighs into strips.
Assemble shawarma: place warm pita on a plate, top with sliced chicken, cooked onions, shredded romaine, chopped tomato, chopped cucumber, and a spoonful of yogurt sauce.