Preheat an outdoor grill for medium-high heat and lightly oil the grate.

Whisk oil, lemon juice, hot sauce, and thyme in a small saucepan. Season with salt and pepper. Arrange chicken and onion slices on a plate. Spoon 2 tablespoons of the dressing over the chicken and onions; turn to coat. Let stand for 5 minutes.

Grill chicken and onion slices on the preheated grill, turning once, until chicken is cooked through and the juices run clear, about 8 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Set the saucepan with the remaining dressing at the edge of the grill to warm.

Transfer chicken and onion slices to a place. Cut chicken crosswise into thin slices. Separate onion slices into rings.

Place lettuce in a large bowl; add chicken and onion. Add warm dressing and toss to coat. Season with salt and pepper. Sprinkle with feta cheese and serve.