Whisk together the creme fraiche, parsley, chives, and tarragon in a bowl; season with salt and pepper. Cover and refrigerate for 30 minutes.
Preheat the oven to 400 degrees F.
Divide the dough in half. Roll each piece into a very thin 10-inch rectangle. Brush with oil, place on baking sheets, and bake until lightly golden brown, about 10 minutes.
Heat 1 tablespoon of the oil and 1 tablespoon of the butter in a small saute pan over medium heat. Add the onion and sugar and cook slowly, stirring occasionally, until caramelized, 30 minutes. Season with salt and pepper.
Heat the remaining 1 tablespoon oil in a medium saute pan over medium heat. Add the bacon and cook until golden brown and crisp, about 10 minutes. Remove with a slotted spoon to a plate lined with paper towels. Divide the creme fraiche between the pizzas, spreading it out.
Divide the cheddar on top and then add the onion and bacon. Bake until the cheese has melted, about 5 minutes. Meanwhile, heat the remaining 3 tablespoons butter in a medium nonstick saute pan over medium heat. Add the chile and cook until soft, about 2 minutes. Whisk the eggs in a bowl until smooth; season with salt and pepper. Pour the eggs into the pan and stir constantly with a rubber spatula until soft curds form. Add the goat cheese. Divide the eggs between the two pizzas and garnish with more herbs.
Recipes reprinted from Brunch @ Bobby's by Bobby Flay. Published by Clarkson Potter Publishers.