Heat an oven-proof 10-inch nonstick skillet over medium-high heat. Add the olive oil and onions and reduce the heat to medium. Cook, stirring often, until the onions are a dark golden-brown, about 15 minutes. Season with 1/2 teaspoon kosher salt. Stir in the balsamic and sprinkle the oregano over the top. Turn off the heat and spread the onions evenly over the bottom of the pan, using a rubber spatula.
Place the ricotta in a small bowl. Season with 1/8 teaspoon each kosher salt and black pepper. Dollop the ricotta over the onions in 8 to 10 places. Using a fork, prick the puff pasty a few times evenly. Place the puff pastry over the top of the filling, pressing down gently to make sure the dough is touching the ricotta and onions.
Bake until the pastry is puffed and golden brown, 35 to 40 minutes. Remove from the oven and cover with a plate slightly larger than the skillet. Invert the crostata onto the plate, replacing any onions that may have stuck to the pan. Set aside to cool slightly.
Meanwhile, in a small bowl, dress the arugula with the lemon juice. Sprinkle the arugula over the cooled crostata and nestle the tomatoes into the onions. Slice into wedges or squares and serve.