Preheat the oven to 400 degrees . In a small bowl, cover the figs with boiling water and steep until softened, about 15 minutes; drain and finely chop.
Meanwhile, in a large skillet, melt the butter over medium heat. Add the onions and thyme and cook, stirring occasionally, until softened and browned, about 15 minutes; let cool slightly.
Arrange the puff pastry on a baking sheet. Using a fork, pierce a 1/4-inch edge around the pastry. In a small bowl, beat together 1 egg and 2 teaspoons water. Brush the pastry with the egg wash; freeze until ready to use.
In a medium bowl, whisk together the ricotta and remaining egg. Stir in the figs; season with salt and pepper.
Spread the ricotta mixture on the puff pastry in an even layer, leaving a border. Top with the ham and onions. Bake until golden, about 20 minutes. Lower the heat to 350 degrees and bake until the bottom of the puff pastry is golden, 5 to 10 minutes.
