With your hands, lightly combine flour and butter until it resembles large chunks. Add water and extract. Quickly mix with hands until dough starts to form. Large butter flecks should remain visible.
Dust the working surface with flour. Using a rolling pin, roll out dough to 1/8-inch-thick circle. Transfer to a baking sheet and refrigerate until the next step is done.
Cut pears in half lengthwise. Remove cores, stems, and seeds. Slice pears thinly at a diagonal, keeping slices together in stacks.
To make frangipane, whisk egg white and sugar in a bowl until frothy and smooth, about 1 minute. Add almonds, melted butter, and almond extract. Whisk to incorporate.
Preheat oven to 400 F and remove dough from refrigerator. Spread frangipane evenly over dough. Fan out pear stacks, leaving a 2-inch border around dough edge. Fold border over pears, overlapping where necessary and pressing gently to adhere folds.
Bake 50 to 60 minutes, or until crust is lightly browned and crispy. Serve warm or at room temperature.