Season the duck legs with salt. Coat a large wide pot lightly with olive oil. Lay the duck legs skin side down, in a single layer, in the pan.
Cook the duck legs low and slow over low heat to render the fat. This process will take a while, don't rush it. It is really important to render as much fat out of the duck legs as possible. Once the pan is full of fat, turn up the heat and brown the duck legs on both sides.
Preheat the oven to 400 degrees F.
Once the legs have rendered and are nice and brown, remove them from the pan and reserve. Add the onions and season with salt. Stir to coat with the duck fat. Cover and cook over medium heat for 15 to 20 minutes. Once the onions are really nice and wilted, remove the lid and cook for another 15 minutes or until the onions start to caramelize. Taste to make sure they are very well seasoned, add salt if needed. Return the duck legs to the pan, snuggle them in with all the onions and add the wine, thyme and bay leaves.
Cover the pan and place in the preheated oven. Braise the duck in the oven for about 90 minutes. Stir the onion and duck every 30 minutes to be sure that the onions are not burning. When done, the duck should be incredibly flavorful, tender and falling off the bone.
Remove the tough lower stems from the dandelion greens, wash and spin them dry. Cut them into 1 -inch lengths. Place the greens in a large bowl with red wine vinegar. Season with salt and some of the warm duck fat. Strain some the onions out of the duck fat and toss them in with the greens.
Place a large mound of the dressed dandelion greens on each serving plate, lay a duck leg on the greens and top with a little more of the onions.