Put the sliced onion in a small bowl and add enough cold water to cover.
Trim off any exposed tendon ends from the wide tips of the tenders, if necessary. Season the tenders with 3/4 tsp. salt and 1/2 tsp. pepper. Heat 2 tsp. of the oil in a 12-inch nonstick skillet over medium-high heat until hot. Cook the tenders until well browned on both sides and just cooked through, about 3 minutes per side. Transfer the chicken to a cutting board and let rest while you make the dressing.
In a large bowl, combine the mint, lemon juice, garlic, lemon zest, and 1/4 tsp. each salt and pepper. Slowly whisk in the remaining 1/3 cup oil. Set aside 2 Tbs. of the vinaigrette in a small bowl.
Drain the onions and press them gently to remove any excess water. Add the onions to the large bowl of vinaigrette, along with the feta, tomatoes, cucumber, and olives. Slice the chicken crosswise into 1/2-inch pieces and add it to the salad. Toss everything to coat. Divide the salad among four serving plates, drizzle with the reserved vinaigrette, and serve immediately.
Serve with warmed pita.