2 cups all-purpose flour
1/4 cup sugar
3/4 teaspoon coarse salt
8 ounces (2 sticks) cold unsalted butter, cut into 1/2-inch pieces
1/4 cup ice water
1/2 pineapple, peeled, quartered and cored
1 1/2 cups strawberries, hulled and quartered
1/4 cup packed light-brown sugar
1/4 cup granulated sugar
1 vanilla bean, split and scraped, pod reserved
2 tablespoons fresh lemon juice
1/2 cup dark rum
To make the crust: Combine flour, sugar, and salt in a large bowl. Add butter and combine until mixture resembles coarse meal. Add ice water and mix until mixture just begins to hold together.
Shape dough into one disk. Wrap in plastic, and refrigerate for at least an hour or up to three days (or freeze for up to 1 month; thaw in refrigerator before using).
Divide crostata dough into 12 parts. Pat each part into a muffin cup in a muffin tin. Trim excess dough. Combine excess dough into a single mass and flatten into a rectangle between two sheets of plastic wrap. Freeze tin and excess dough until firm, about 30 minutes.
To make the filling: Cut each pineapple quarter crosswise into 1/3-inch-thick slices. Combine sugars, vanilla seeds and pod, and lemon juice in a large saute pan. Add pineapple and strawberries and cook over medium heat, stirring until sugars dissolve and mixture becomes saucy, about 3 minutes. Add rum and simmer until pineapple has softened and almost all of the liquid has evaporated, about 20 minutes. Remove from heat, and let cool in pan. Discard vanilla pod.
Remove muffin tin from refrigerator. Divide pineapple-and-strawberry mixture evenly into each cup. Take remaining rectangle of dough and cut into thin strips place four strips crosswise over each cup.
Preheat oven to 400 degrees. Bake for 15 minutes, or until tarts are golden brown around the edge. Let cool in pan for 30 minutes before removing. Unmold and serve immediately.