Whisk together the crème fraiche, parsley, chives, and tarragon in a bowl; season with salt and pepper. Cover and refrigerate for 30 minutes.
Preheat the oven to 400°F.
Divide the dough in half. Roll each piece into a very think 12x10-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 sauté 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 2 tablespoons oil in a medium sauté 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 crème fraiche between the pizzas, spreading it out. Divide the cheddar and sprinkle it on top and then add the onion and bacon. Bake pizzas until the cheese has melted, about 5 minutes.
Meanwhile, heat the remaining 3 tablespoons butter in a medium non-stick sauté 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.