Make the dough: In a large bowl, combine the warm water, yeast, all-purpose and whole wheat flours, and salt. Mix with a rubber spatula until well-combined and the dough comes together into a ball.
Lightly dust a clean surface with flour, then turn the dough out and knead for 1–2 minutes to smooth and encourage gluten development. Shape the dough into a ball.
Drizzle the olive oil in the bowl. Add the dough and turn to coat in the oil. Cover with a kitchen towel and let rise for 1 hour, or until doubled in size.
Caramelize the onions: Heat 2 tablespoons of olive oil in a large pan over medium heat. Add the onion, ½ teaspoon salt, and the pepper. Sauté for 5 minutes, until slightly softened. Add the balsamic vinegar, stir, and reduce the heat to low. Cook for about 25 minutes more, until the onions are tender and caramelized. Remove the pan from the heat and set aside to cool.
Punch down the dough, then divide into 4 pieces. On a lightly floured surface, roll out each piece to an 8-inch (20 cm) round, about ⅛ inch (3 mm) thick.
Heat a grill pan over medium-high heat. Brush each side of the dough lightly with olive oil. One at a time, grill each flatbread for about 4 minutes on each side, until grill marks appear and the dough is cooked through.
Add the arugula to a medium bowl, along with the caramelized onions, remaining ½ tablespoon olive oil, and remaining ¼ teaspoon salt. Toss well.
Top each flatbread with some of the arugula mixture, 6 peach slices, and 1 ounce of crumbled goat cheese. Drizzle with the balsamic glaze.