In a small bowl, combine 4 tablespoons of the oil, the lemon juice, capers, and parsley. Set aside.
Bring a large pot of salted water to a boil. Add the green beans and cook until just tender, 4 to 6 minutes. Drain and run under cold water.
Cut the Feta into 4 slices. Set aside to drain on paper towels.
Heat a large nonstick skillet over medium heat. Season the tomato slices with thyme, ½ teaspoon salt, and ¼ teaspoon pepper. Cook until lightly browned, about 1 minute per side.
Wipe out the skillet and heat the remaining tablespoon of oil over medium heat. Coat the Feta in bread crumbs and cook until browned, about 2 minutes per side.
Arrange the arugula, tomatoes, beans, and Feta on plates and drizzle with the vinaigrette.