Place the potatoes, cauliflower, carrots and parsnips into a large saucepan and cover with cold water by 2 inches. Bring the water to a simmer, salt it generously and cook the vegetables until just tender, about 10 minutes. Drain well, shaking off as much moisture as possible, and then transfer to a large bowl with the butter. Season with salt and pepper and roughly mash, leaving lots of large, coarse chunks.
Meanwhile, cook the bacon in a large heavy-bottomed skillet over medium heat until brown and crisp, about 8 minutes. Transfer to the bowl with the vegetables using a slotted spoon, leaving as much bacon fat in the skillet as you can. Cook the Brussels sprouts in the same skillet over medium heat until wilted and tender, about 5 minutes, and then transfer to the bowl of vegetables. Mix well and divide the vegetables and bacon into 6 even patties. Refrigerate until ready to cook.
Heat 2 tablespoons of the oil in a medium skillet over medium heat. Cook the patties in batches until the bottoms turn golden brown, 4 to 5 minutes, and then flip and cook until the second sides are crisp and golden brown, another 4 to 5 minutes. Transfer to a platter.
Heat 1 tablespoon of the oil in a large nonstick skillet over medium heat. Crack 3 of the eggs into the pan, sprinkle with salt and pepper and cook until the whites are set. Place 1 egg on top of each bubble and squeak cake. Fry the remaining eggs and place on the remaining cakes, and then serve topped with the chives.
Recipe courtesy Young Sun Huh for Cooking Channel