In a slow cooker on low, add the onions, 2 tablespoons oil and a pinch of salt. Cook, stirring every hour or so, until deep golden brown, 8 to 10 hours. Cook uncovered for the last hour.
Preheat the oven to 400 degrees F.
Roll the dough out to fit an 11-by-17-inch baking sheet or a large round pizza pan. Drizzle the remaining 2 tablespoons olive oil over the top and pierce the dough with a fork all over to minimize bubbles. Bake the dough, checking every few minutes to ensure it doesn't burn, until lightly golden in the center, about 15 minutes. Remove from the oven and set aside. (Leave the oven on.)
In a medium bowl combine the Monterey Jack and Parmesan. Add half of the cheese mixture to the mashed potatoes.
Top the dough in an even layer with the mashed potato mixture. Add the caramelized onions, bacon, chile flakes and the remaining cheese. Bake until the cheese is melted and the edges of the crust are golden, 8 to 10 minutes. Top with a drizzle of the yogurt and sprinkle with the chives.