Make marinade: In a small bowl, whisk together ¼ cup oil, juice of 2 lemons, garlic, Italian seasoning, salt, and pepper. To a large ziploc bag, add chicken breasts and marinade and toss to coat. Let marinate for at least 30 minutes and up to 4 hours.
In a large skillet over medium-high heat, heat remaining two tablespoons oil. Add chicken breasts and cook until golden and cooked through, about 8 minutes per side. Remove from pan and let rest a few minutes, then slice.
Make spicy yogurt sauce: In a small bowl, mix together yogurt, hot sauce, remaining lemon juice, and season with paprika and more salt.
Assemble wraps: To a piece of pita, add lettuce, tomato, chicken slices, and drizzle with spicy yogurt sauce. Wrap in a piece of parchment and serve.