Preheat the oven to 425°F.
Toss the cubed squash with 1 tablespoon of the olive oil and a pinch of salt. Spread in a single layer on a sheet pan.
Roast the squash for 20 to 25 minutes, stirring once halfway through, until fork-tender and browned at the edges.
While the squash roasts, heat the remaining olive oil in a large skillet over medium heat. Add the onion and cook for 4 to 5 minutes, until softened.
Add the garlic and smoked paprika and cook for 30 seconds until fragrant.
Add the roasted squash to the skillet along with the thawed edamame. Toss together and cook for 2 to 3 minutes, just until the edamame is heated through.
Season with salt and pepper to taste.
While the hash finishes, fry or poach the eggs to your preference.
Divide the hash among plates, top each with an egg, and garnish with fresh parsley.