Preheat the oven to 400 degrees with rack in center. In the bowl of a food processor fitted with the steel blade, puree ricotta until smooth. Add one egg yolk, and 1/2 tablespoon olive oil. Pulse to combine. Transfer to a small bowl, and fold in creme fraiche, 1/8 teaspoon salt, and a pinch of pepper. Set aside.
Heat 1 tablespoon oil in a large skillet over high heat. Add half of the mushrooms, 1/4 teaspoon salt, a pinch of pepper, and 1 teaspoon thyme. Cook, stirring occasionally, until all the released liquid has evaporated, 3 to 4 minutes. Stir in 1 tablespoon butter, and continue to cook until mushrooms are tender and slightly crisp, about 2 minutes more. Transfer to a medium bowl. Repeat process with remaining mushrooms.
While the mushrooms are cooking, prepare the onions: Cut the white bulbous part of the onions into 1/2-inch wedges. Slice enough of the greens across on the diagonal to equal 1/2 cup; set aside. Discard the remaining greens.
Heat the remaining 2 tablespoons butter over low heat. Add the onion wedges, remaining 1 teaspoon thyme, 1/4 teaspoon salt, and a pinch of pepper. Cook until soft and just beginning to color, about 2 minutes. Be sure not to stir onions too much during cooking so they hold their shape. Remove from heat, and let stand in pan until cool to the touch.
Make an egg wash: In a small bowl, whisk together the remaining egg yolk and 1 tablespoon water. Lightly dust a work surface with flour. Roll out puff pastry 1/4 inch thick. Using a pastry wheel, cut into an 8-by-12-inch rectangle, reserve scraps for another use. Transfer to a baking sheet. Score the edge of the tart by running a sharp knife around the perimeter of the pastry 1/4 inch from edge, cutting only halfway through dough. Brush edges of tart with egg wash.
Spread ricotta mixture evenly within the borders of the scoring. Scatter half of the mushrooms over the ricotta. Lay the Cantal slices evenly over the mushrooms. Gently combine the remaining mushrooms with the cooled onions and the sliced onion greens. Scatter this mixture evenly over the cheese. Can be made ahead up to this point and refrigerated for up to 1 day.
Bake until cheese is bubbly and crust is golden brown, about 24 minutes, turning once after 12 minutes. Remove from oven, let rest for 1 to 2 minutes. Cut into 6 slices. Serve with parsley salad.