To make crust, sift together dry ingredients and pulse in food processor to distribute evenly. Add butter and pulse together until butter flakes into pea-size pieces, 1 to 2 minutes. Gradually add ice water and pulse until dough just holds together.
Turn out dough onto lightly floured surface. For final blending, smear dough a few spoonfuls at a time with heel of your hand. Gather dough with scraper and shape into round disc. Wrap dough in plastic wrap and refrigerate at least 1 hour or overnight.
Preheat oven to 375ºF. Grease 9-inch pie plate or rectangular baking pan with butter.
Combine sugars, cinnamon, and salt in small bowl. Sprinkle bottom of greased pan evenly with half sugar mixture. Layer plantains cut-side up in pie plate. Drizzle plantains with the rum and top with remaining sugar mixture. Pour melted butter over plantains.
On lightly floured surface, roll out dough to round, ⅛ inch thick. Dock dough so it rises evenly, using fork to make small incisions. Lay dough over plantains (so they are entirely covered) and fold corners back on themselves.
Place in preheated oven and bake until crust is lightly browned, 45 to 60 minutes.
Remove tart from oven and unmold while still warm. Set serving dish over pie plate and carefully flip, allowing caramel to drip down over tart. Serve with scoop of ice cream or whipped cream.