In a medium skillet, heat the 1/4 cup of olive oil. Add the mushrooms and cook over moderately high heat, stirring occasionally, until lightly browned, about 5 minutes. Add the shallot and garlic and cook until fragrant, about 1 minute. Add the wine and cook until evaporated, about 1 minute. Season with salt and pepper.
Light a grill or preheat a grill pan and lightly oil a rimless baking sheet. On the baking sheet, stretch each ball of pizza dough out to a 12-inch round; brush with extra-virgin olive oil. Carefully slide the dough rounds onto the grill and cook over moderately high heat, shifting the rounds to prevent scorching, until lightly browned on the bottom, about 2 minutes. Flip the rounds and spread 1/2 cup of the fresh ricotta cheese over each one. Scatter the sautéed mushrooms and herbs over the flatbreads and grill until heated through, about 2 minutes.
Transfer the flatbreads to a work surface, drizzle lightly with olive oil and season with salt and pepper. Cut into wedges and serve right away.