Roast potatoes and beets: Preheat oven to 400 degrees with one rack in the center and one rack in the bottom third. In a small roasting pan, toss potatoes with 1 tablespoon oil, the thyme, salt, and pepper, and cover with aluminum foil. In a second small roasting pan, combine beets and the remaining tablespoon oil. Season with salt and pepper, add 1/2 cup water, and cover with aluminum foil. Transfer potatoes and beets to oven, and bake until tender when pierced with the tip of a knife, 35 to 45 minutes for the potatoes and 45 minutes to 1 hour for the beets. Remove from oven, and let stand until cool; set aside.
Prepare salmon: Reduce oven temperature to 250 degrees and place a roasting pan filled with water on bottom rack. Line a rimmed baking sheet with parchment paper and lightly oil parchment; set aside. Chop parsley, shallots, dill, tarragon, and lemon zest together until very fine. Transfer to a small bowl and stir in oil; set aside.
Season skin side of fish with 2 teaspoons coarse salt and a pinch of pepper. Place skin side down on prepared baking sheet. Season with fleur de sel and more pepper. Coat fish with herb mixture. Transfer to oven, and bake 30 minutes for rare to medium rare. Remove from oven, and let stand until room temperature.
To assemble: Cut potatoes in half. Toss with 2 teaspoons mustard vinaigrette and 1/2 teaspoon salt. Peel beets, and cut into wedges. Place in a small bowl. Season with 1/8 teaspoon salt, a pinch of pepper, the olive oil, and lemon juice. Toss to combine. Cut eggs in half lengthwise, and season with salt and pepper.
In a medium bowl, toss dandelion greens with 3 tablespoons vinaigrette, a pinch of salt, and a pinch of pepper. Arrange greens on a platter. Top with potatoes, beets, and eggs. Top with salmon. Alternatively, divide greens among six plates. Place 3 potato halves on each salad. Spoon beets around each plate. Flake salmon and lay over greens in center of plate. Garnish each plate with half an egg.