Rinse and drain fennel. Trim off tops; discard tough stalks and chop enough feathery greens to make 1/4 cup. Save remaining greens for other uses or discard. Trim and discard root ends and bruised areas from fennel heads. Cut fennel crosswise into 1/16- to 1/8-inch-thick slices. Combine sliced fennel and chopped fennel leaves in a wide, shallow bowl.
Peel and stem persimmons. Cut fruit crosswise into 1/8-inch-thick slices and put in another bowl. Add lemon juice and oil; mix gently. Holding fruit back, pour liquid over fennel. Mix fennel with dressing.
Arrange the persimmon slices over the fennel; sprinkle with cheese. Spoon salad onto plates and add salt and pepper to taste.