Place almonds in a skillet over medium heat. Cook, shaking skillet often, until nuts are lightly toasted, 5 to 7 minutes. Transfer to a bowl.
Preheat oven to 375°F; line a large baking sheet with parchment. Place one sheet of puff pastry on a lightly floured surface. Brush it lightly with water; place second sheet on top. Using a rolling pin, roll pastry sheets together into a 12-inch square, then trim into a 12-inch circle. Transfer to lined baking sheet; refrigerate.
In a bowl, using an electric mixer on medium speed, beat almond paste, 3 Tbsp. sugar and 1 egg until smooth. Beat in butter and almond extract until smooth. Stir in flour. Spread mixture evenly over pastry, leaving a 1-inch border. Fan pears in a circle over filling. Scatter cherries on top. Fold pastry edges over filling, pleating as you go, leaving center of pears exposed. The pastry should tightly enclose pears and filling.
Beat remaining egg with 1 tsp. water. Brush mixture over pastry edges. Sprinkle remaining sugar over fruit. Bake until pastry is golden, about 45 minutes. Sprinkle almonds over fruit and transfer tart to a wire rack to cool. Serve at room temperature.