Cut 1/2 cup butter into small pieces and put in a bowl. Add 1/2 cup granulated sugar and stir to combine, then beat with a mixer on high speed until well blended, about 2 minutes. Add orange peel, almond extract, salt, and eggs. Stir to mix, then beat on high speed until well blended and smooth, 3 to 4 minutes. Add 1 cup flour and stir to mix, then beat on high speed to blend.
Butter and flour an 11- to 11 1/2-inch tart pan with removable bottom. Scrape batter into pan and spread level.
Rinse plums, cut in half, and pit; cut fruit into 1/4-inch-thick slices. Overlap all the slices on batter; sprinkle with remaining 2 tablespoons granulated sugar.
Bake on the middle rack of a 375° regular or convection oven until batter that pops up around fruit is lightly browned and tart begins to pull away from pan sides, 35 to 45 minutes. Let cool about 10 minutes. Dust with powdered sugar (on the cut fruit, the sugar melts quickly).
Remove pan rim (if tart sticks, use a small knife to release it from the rim) and set on a plate. Serve hot, warm, or cool.