Trim off and discard the fennel stalks. Halve and core the bulbs. Using a handheld slicer, cut into paper-thin slices.
In a large salad bowl, toss the fennel, both parsleys and the lettuce.
In a small bowl, whisk the vinegar, mustard, sugar, 1/2 tsp. salt and 1/4 tsp. pepper. Whisk in the EVOO, then the chives. Drizzle the dressing over the greens and toss.