Place the tomato slices on a baking rack and sprinkle with salt. Set aside.
In a saute pan, heat the olive oil over medium-low heat and cook the onions until golden brown, about 15 minutes. Set aside.
Meanwhile, cut an 8-inch circle out of the puff pastry and press into an 8-inch pie pan. Poke the bottom of the pastry with a fork evenly throughout. Reserve the trimmings. Brush the egg white along the border of the circle and press the scraps of puff pastry along the side of the pan to create a crust. Scatter the goat cheese crumbles on top as the filling of the pie and then the caramelized onions. Layer the tomatoes on top in decorative circles. Season with salt and pepper.
Bake for 20 minutes, then cover the edges of the pastry and bake until the veggies are soft and beginning to brown, 20 more minutes. Garnish with fresh basil.