2 cups caramelized onions
2 tablespoons plus 1 teaspoon finely chopped fresh parsley leaves, divided
1 1/2 teaspoons finely chopped fresh thyme leaves, divided
1/3 cup olive oil
10 (14x18-inch) phyllo sheets (about 8 ounces), defrosted if frozen
Freshly ground black pepper
1 (5.2-ounce) box garlic and herb Boursin cheese
2 cups arugula
1 teaspoon freshly squeezed lemon juice
Arrange a rack in the middle of the oven and heat to 400°F. Line a standard 18x13-inch rimmed baking sheet with parchment paper; set aside.
Place the onions, 2 tablespoons of the parsley, and 1 teaspoon of the thyme in a medium bowl and mix to combine; set aside. Add the remaining 1 teaspoon of parsley and 1/2 teaspoon of thyme to the olive oil and stir to combine; set aside.
Lay 1 sheet of phyllo on the prepared baking sheet. Keep the remaining phyllo covered with plastic wrap, wax paper, or a damp kitchen towel. Brush the phyllo with a thin layer of the herb oil. Repeat with the layers of phyllo, brushing each layer with oil, until all the phyllo is used up, making sure to brush the top layer of phyllo with the oil.
Season the surface of the phyllo with salt and pepper. Carefully roll about 3/4-inch of each side in on itself toward the center to form the outer rim of the flatbread. Spread the onions evenly over the phyllo. Crumble the Boursin evenly over the onions.
Bake until the edges are golden-brown and crisp, about 30 minutes. Transfer the tart to a cutting board.
Place the arugula and lemon juice in a medium bowl, season with salt and pepper, and toss gently with your hands to combine. Sprinkle the tart evenly with the dressed arugula, cut into 12 pieces, and serve warm or at room temperature.