Combine the flour, 1 tablespoon sugar, and the salt in the bowl of a food processor fitted with the S-blade. Blitz until combined.
Add the butter and pulse until the butter is the size of small peas and is coated in flour.
With the processor running, slowly add the water, until the dough begins to clump and form a ball around the blade. You may not need all of the water. Wrap the dough in plastic and chill for at least 1 hour and up to 24 hours.
Preheat the grill for medium indirect heat (375 to 400°F | 190 to 200°C), ideally with heat on both sides of the indirect zone. Butter a 12-inch cast iron skillet well.
While the grill is preheating, combine the sliced plums and 2 tablespoons of sugar in a medium bowl. Let sit for 10 minutes.
While the plums are macerating, roll the dough out on a floured surface to a circle about 13-inches (33 cm) in diameter.
Use a slotted spoon to transfer the plums to a separate medium bowl, leaving any juices behind. Stir in the remaining 5 tablespoons sugar, cornstarch, cinnamon, cardamom, and a pinch of salt.
Spoon the plums into the center of the dough, arranging decoratively if desired. Fold the edges of the dough over the edge of the filling, leaving an open space in the center.
Place the skillet over indirect heat on the grill and cook, covered, until the filling is bubbling and the crust is golden brown, about 50 minutes. Cool completely before serving with ice cream or whipped cream, if desired.