In a food processor pulse the flour, sugar, lemon zest, and salt until blended. Add the butter and pulse again, just until the mixture resembles coarse crumbs. Add water, 1 tablespoon at a time, pulsing just until the mixture forms a ball (do not over mix). Gather the dough together and form into a disk. Wrap in plastic wrap and refrigerate for 1 to 2 hours.
Preheat the grill for indirect cooking over medium-low heat (about 300°F), and preheat a pizza stone for at least 15 minutes, following manufacturer’s instructions.
In a bowl combine the strawberries, cornstarch, sugar, vanilla, and lemon juice. Let stand for 5 minutes while you roll out the dough.
Cut a piece of heavy-duty aluminum foil into an 18-inch square and set on an inverted sheet pan. Brush the foil with oil.
Lightly flour a work surface, a rolling pin, and the dough. Roll the dough into a 12-inch circle. Fold the dough onto the rolling pin and transfer to the foil.
Put the strawberry mixture and any juices on the dough, leaving a 2-inch border. In a microwave-safe dish, warm the preserves for 10 to 20 seconds. Brush on the strawberries. Top with the almonds. Fold over and pleat the edges of the dough (leaving an opening in the center) and brush the dough with half-and-half. Fold up the edges of the aluminum foil around the crostata to create a 2-inch wall to prevent the juices from spilling over. Gently transfer the foil-stabilized crostata from the sheet pan to the preheated pizza stone. Grill over indirect medium-low heat, with the lid closed, until the strawberries are bubbling and the crust is golden brown, 45 minutes to 1 hour.
Lifting the foil edges, carefully remove the crostata from the pizza stone. Cool for at least 15 minutes before serving.