Persimmon trees have long been prized by the nation's gardeners for ornamental landscaping, but the fruit's tart, mellow-sweet flavor is also a rare treat for cooks.
Preheat oven to 425 degrees. Wrap beets in parchment-lined foil (each color in a separate packet). Place on a rimmed baking sheet and roast until knife-tender, about 45 minutes. When cool enough to handle, rub beets with paper towels to remove skins. Cut into 1/4-inch-thick rounds.
Remove peels and pith from citrus fruits with a sharp knife. Working over a small bowl to catch juices, carefully cut between membranes to remove segments. Squeeze juice from membranes into bowl.
Arrange beets, citrus segments, persimmons, endive, and watercress on a platter. In a bowl, whisk together 3 tablespoons reserved citrus juices and vinegar. Season with salt and pepper. Slowly whisk in oil. Drizzle vinaigrette over salad just before serving.