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.