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.
Sauté the vegetables: Heat the olive oil in a large skillet over medium-high heat. Once the oil is shimmering, add the mushrooms. Sauté for 3–5 minutes, until the mushrooms are beginning to brown lightly. Reduce the heat to medium and add the red onion, salt, and pepper. Cook for another 3–5 minutes, until slightly softened. Add the garlic and bell peppers and sauté for 2 minutes more, until the peppers are just beginning to soften. Remove the pan from the heat.
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.
Turn the broiler to high. Line 2 baking sheets with parchment paper and place 2 flatbreads on each pan.
Evenly divide the sautéed vegetables between the flatbreads. Scatter the mozzarella over the top.
Broil the flatbreads for 2–3 minutes, watching carefully to make sure they don’t burn, until the cheese has melted and begins to brown.