How to Make It
Warm oil in a large skillet over medium-high heat. Add leek and celery; cook, stirring, until softened but not browned, about 5 minutes. Add carrots, potato, parsnip, turnip and beet. Season with salt and pepper. Pour in 1/4 cup water; loosely cover. Cook, stirring, until vegetables are soft but not mushy and most of liquid has been absorbed, 10 to 12 minutes. Season with salt and pepper, if desired.
Melt butter in a large nonstick skillet over medium-high heat. Crack eggs into skillet and cook until desired doneness. Season with salt and pepper. Divide hash among 4 dishes and top each with an egg.