Line a baking pan with parchment. In a large skillet, melt the butter over medium-low heat. Add the pears and lemon juice. Sprinkle the pears with 1 1/2 tablespoons sugar and the ginger, cover and cook until crisp-tender, about 7 minutes. Uncover and cook until tender and syrupy, 3 to 4 minutes; let cool completely.
Meanwhile, using a food processor, finely grind together 1/4 cup almonds, the remaining 1 1/2 tablespoons sugar and the vanilla.
Dust the prepared pan with flour. Place the puff pastry on the pan and roll out into an 11-inch square, then cut into a 10-inch round. Top with the ground almond mixture, leaving a 1/2-inch border. Arrange all but 1 pear quarter in a circle on the pastry, narrow ends toward the center. Cut the remaining piece in half and arrange in the center of the pastry. Top the pears with the remaining 2 tablespoons almonds. Refrigerate for 30 minutes or up to 12 hours.
Preheat the oven to 400 degrees . Bake the tart until the edge is puffed and golden, about 15 minutes. Lower the heat to 375 degrees and bake until cooked through, about 10 minutes.