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Line a large baking sheet with paper towels. Set the sliced tomatoes on top in a single layer. In a small bowl, combine 1 teaspoon salt and 1 teaspoon pepper. Use it to season the tomatoes evenly on both sides, cover with another paper towel and let sit, 15 minutes, allowing the salt to draw moisture out of the tomatoes.
As the tomatoes sit, heat the oven to 400 degrees and set a rack in the middle of the oven. Working on a large sheet of parchment paper, roll out the puff pastry into a 9-by-11-inch rectangle, trimming any uneven edges. Prick the inside with a fork every few inches, leaving a 1/2-inch border. Using a pastry brush, coat the center of the puff pastry with the crème fraîche, leaving the border unbrushed.
Working within the border, layer the tomatoes and red onion on top of the tart, allowing them to overlap slightly. Transfer to a sheet pan and bake, rotating halfway through, until puff pastry is browned and puffed, 30 to 35 minutes.
Sprinkle tomatoes with Aleppo pepper, if using. Dollop with fresh ricotta. Thin pesto with olive oil until it reaches the proper consistency for drizzling; drizzle on top of tart to taste. Top with basil, if using, and serve warm or at room temperature.