COMBINE the mayonnaise, vinegar, chives, garlic, sugar, salt, Worcestershire, and hot sauce in a food processor or electric blender. Pulse several times to blend. With the motor running, slowly add the oil in a steady stream. Fold in the parsley, cover and chill.
ARRANGE the salad greens on salad plates. Arrange tomato and cucumber slices over the greens, and drizzle with the dressing and serve.