Preheat a gas or charcoal grill with a cover. (The pizzettes can also be baked in the oven. Place an oven rack in the upper 1/3 of the oven and preheat to 400 degrees F. Bake the pizzettes on a cornmeal-dusted baking sheet until the crust is golden, 12 to 14 minutes.)
Dust a baking sheet with cornmeal. Set aside. Using a mandoline slicer, slice the apples and beets into 1/16 to 1/8-inch thick slices. Set aside.
On a lightly floured work surface, roll out the dough into a 13-inch diameter circle. Using a 4-inch round cookie cutter, cut out 6 circles of dough and transfer to the prepared baking sheet. Divide the fontina and sprinkle on top of each circle of dough. Divide the apple and beet slices and place on top. Add the goat cheese and sprinkle with herbes de Provence. Drizzle with olive oil and place on the grill over low, indirect heat. Cover the grill and cook until the crust is crisp and golden and the cheese begins to melt, 8 to 10 minutes.
*For easier shredding, freeze the fontina cheese for 15 minutes.