Put the potatoes in a saucepan, add enough cold water to cover by 2 inches, salt the water and bring to a boil over high heat. Boil until cooked three-quarters of the way through, or not quite tender when pierced with a knife, 15-20 minutes. Drain, let cool slightly, peel, and then grate using the large holes of a box grater.
Preheat the oven to 425°F.
Heat 2 tablespoons of the oil in a large sauté pan over medium heat. Add the onion, season with salt and pepper, and cook, stirring occasionally, until golden brown and caramelized, about 30 minutes.
Heat the remaining 2 tablespoons oil in a large, nonstick sauté pan over high heat. Add the potatoes and cook until golden brown. Stir in the caramelized onions, poblano and 1 tablespoon of the ancho powder. Season with salt and pepper, and cook until just warmed through, about 2 minutes.
Cook the bacon in a skillet over medium heat until browned and crisp, about 8 minutes. Scoop out and transfer to a plate lined with paper towels to drain.
Lay out 8 of the tortillas on a work surface. Divide the hash browns, bacon and cheese among the tortillas. Stack half of the tortillas on top of the remaining topped ones to create 4 double stacks. Top each with one of the remaining 4 tortillas. Brush the tops with the little oil and sprinkle with the remaining 2 teaspoons ancho powder.
Transfer the quesadillas to a large baking sheet and bake until golden brown and the cheese has melted, 8-10 minutes.
When the quesadillas are nearly ready, melt the butter in a large, nonstick sauté pan over medium heat. Carefully crack the eggs into the pan, season with salt and pepper, and cook until the whites are set but the yolks are still runny, about 2 minutes.
Top each quesadilla with a fried egg, some of the tomato salsa and sprinkling of chives.
For the Fresh Tomato Salsa
: Combine all of the ingredients in a bowl and let sit at room temperature for a least 15 minutes before serving.