To make the dressing, combine the oil and lemon juice in a small jar and shake vigorously. Mix in the capers and parsley; set aside.
Heat a large pot of lightly salted water to boiling. Drop in the green beans and cook, uncovered, 5 minutes or until crisp-tender. Drain, then rinse immediately with cold water. Cut the Feta into 4 slices and set aside to drain on paper towels. Heat a large nonstick skillet over medium heat. Sprinkle the tomato slices with the thyme, salt, and a few grinds of pepper. Place the tomato slices in the skillet and cook until lightly browned, about 1 minute. Flip and brown the other side. Arrange the arugula on 4 salad plates. Place a few tomato slices on each plate along with some of the green beans. Coat the Feta in the bread crumbs. Add the oil to the skillet and brown the cheese on both sides, about 2 minutes per side. Add to the salad plates. Drizzle the dressing over the salads.