Combine first 5 ingredients in a large bowl; stir with a whisk. Set aside.
Punch Flatbread Dough down; cover and let rest 5 minutes. Divide dough into 2 equal portions. Working with one portion at a time (cover remaining dough), roll each half into a 12-inch circle on a floured surface. Place each dough circle on a baking sheet sprinkled with 1 tablespoon cornmeal. Brush each portion with 1 teaspoon oil, leaving a 1/2-inch border. Pierce entire surface liberally with a fork. Bake at 350° for 10 minutes or until crisp.
Add arugula, plums, and almonds to vinegar mixture; toss gently to coat.
Remove flatbreads to cutting board; top each with half of arugula mixture and 1/4 cup goat cheese. Cut each piadina in half.
I really love this meal, but haven't made it as often I would like because of the time required for the dough. I use mache instead of arugula. I love the texture that the cornmeal gives the flatbread. It's suprisingly hearty.