Slice the plums in half, and remove pits. Transfer plum halves to a mixing bowl. Add the salt, pepper, and thyme.
Heat the olive oil in a large skillet set over medium heat. Add the plum halves, cut-sides down, and cook, shaking skillet often, until plums release their juices but still hold together, 5 to 6 minutes. Add the wine and balsamic vinegar; cook until the juice slightly thickens, about 2 minutes more.
Arrange arugula in a mound on a serving platter. Drape the prosciutto over the arugula. Spoon the plums and sauce over the arugula and the prosciutto, and serve.