Use a mandoline or very sharp knife to shave the carrots lengthwise into ribbons. Cut off the stalks and fronds from the fennel bulbs if still attached. Shave the fennel lengthwise like the carrots.
Put the carrots and fennel in a bowl and toss with the olive oil, lemon juice, and 1 tablespoon vinegar. Let stand for 20 minutes to soften.
Add the red onion. Gently mix in watercress and tangerines. Season with salt and pepper, toss well, taste, and add more vinegar if needed. Serve immediately.
Michael's Notes: If your tangerines are small, you can leave the sections intact. If the tangerines are large, or if you want to use oranges, you should remove each citrus section from its membrane. To do so, cut a slice off both ends of each fruit so it will stand upright. Stand each fruit on a cutting surface and, using a sharp knife, remove all the peel and white pith by slicing from top to bottom all the way around the fruit, following its contours. Cut the sections away from their membranes.