In a large bowl, whisk together vinegar, oil, and thyme; season with salt and pepper. Add the arugula, red-leaf lettuce, frisee, strawberries, and walnuts, if using. Season with salt and pepper; toss to combine.
On each of 4 individual salad plates, arrange 3 large endive leaves in a 3-pointed star with tips facing out. Then place equal mounds of salad in the center of each plate. Scatter goat cheese on top, if desired. Serve immediately.