1. Prepare the dressing: In a small bowl, blend all the ingredients and set aside.
2. Make the salad: In a medium saucepan, cover the potatoes with cold water and bring to a boil over high heat. Boil until tender, about 15 minutes. Drain and let cool slightly. Peel the potatoes and slice them 1/4 inch thick.
3. In a large nonreactive skillet, warm 1 1/2 tablespoons of the olive oil over high heat. When the oil is hot, add the mushrooms in an even layer and cook undisturbed until browned on the bottom, about 3 minutes. Reduce the heat to moderately high, season with salt and pepper and sauté until tender, about 4 minutes. Add the butter, garlic and shallot, reduce the heat to moderately low and cook, stirring, until the shallot is tender, about 4 minutes. Scrape the mushrooms onto a plate.
4. Set the skillet over high heat. Add the remaining 1 tablespoon oil and when it is very hot, add the potatoes in an even layer. Season with salt and pepper and cook undisturbed until brown on the bottom, about 4 minutes. Turn the potatoes and stir in the mushrooms. Add the wine and cook, stirring, until absorbed. Add the lemon juice, thyme, and rosemary and remove from the heat. Season with salt and pepper.
5. Warm a nonstick medium skillet over moderate heat for 1 minute. Measure out 1/3 cup of the grated cheese and sprinkle it into the skillet in a 6-inch round. Cook the cheese until it melts and becomes a lightly browned disk, about 4 minutes. Loosen the frico from the skillet and slide it onto a plate to cool; it will crisp up as it cools. Repeat to make 7 more disks.
6. Set 4 frico disks on a work surface. Reheat the potatoes and mushrooms in the skillet and spoon the mixture evenly onto the disks. Drizzle about 1/2 tablespoon of the dressing over each serving and top with a second frico disk. With a large sharp knife, quarter each filled frico.
7. In a large bowl, toss the watercress and radicchio thoroughly with the remaining dressing. Mound the greens on 4 large plates and arrange a quartered frico on top of each salad. Serve immediately with a knife and fork.
Make Ahead: The potato and mushroom filling and the frico disks can stand at room temperature for up to 6 hours.
Wine Recommendation: A simple but characterful white is the best choice. Look for a Pinot Blanc from Alsace, such as the 1994 Lucien Albrecht Pinot Blanc d'Alsace, or an Oregon Pinot Gris, such as the 1994 King Estate.