Whisk together flour, 1 Tbsp. sugar, baking powder, baking soda and salt. Cut 4 Tbsp. butter into little pieces, add to flour mixture, and cut in butter with fingertips or two knives until mixture is mealy. Stir in half-and-half until dough forms. Mold dough into a brick shape, place in plastic wrap and chill while you prepare topping.
Halve and pit plums and then cut each half into 3 wedges. Melt remaining 2 Tbsp. butter in a small saucepan or in microwave.
Preheat oven to 450°F. Line a baking sheet with parchment paper.
Roll out dough on a lightly floured surface to a 12-by-5-inch rectangle. Trim edges with a large, sharp knife to neaten. Then transfer dough base to parchment-paper-lined baking sheet. Brush dough with half of melted butter. Arrange plum wedges horizontally on dough in 5 rows. Dab with remaining melted butter and sprinkle with remaining 1 Tbsp. sugar.
Bake until crust is golden and fruit is softened, about 20 minutes. Let tart cool for at least 10 minutes on baking sheet. (If any fruit shifts during baking, rearrange it on tart with a fork while still warm.) Serve with a scoop of vanilla ice cream, if desired.