Blend all ingredients in processor until almost smooth. Season with salt and pepper.
Preheat oven to 400°F. Spray large rimmed baking sheet with nonstick spray. Toss potatoes and next 5 ingredients on prepared sheet; spread in single layer. Sprinkle generously with salt and pepper. Roast vegetables until tender, stirring and turning occasionally, about 45 minutes. Stir in garlic; roast 5 minutes longer. Mix in green onions. Fill large skillet halfway with generously salted water; bring to boil. Reduce heat to maintain steady simmer. Crack eggs, 1 at a time, into custard cup, then slide eggs into simmering water. Poach eggs until softly set, about 3 minutes.
Divide hash among 4 plates. Using slotted spoon, top each serving with 1 poached egg. Drizzle with pesto.