Place the potatoes in a medium saucepan and add just enough cold water to cover. Place over high heat and season with salt, stirring to dissolve. Bring to a boil, then immediately remove from the heat, drain, and set aside.
Preheat the oven to 400 degrees F.
In a 12-inch ovenproof nonstick skillet, heat the butter and oil over medium-high heat. Add the potatoes and spread out in a single layer. Cook, undisturbed, until golden brown on the bottom, about 8 minutes. Toss the potatoes, spread in a single layer again, and cook, stirring only occasionally, until most of the potatoes are golden brown and tender, about 4 minutes longer. Add the onion, bell pepper, 1/2 teaspoon salt, and 1/4 teaspoon pepper and stir well to combine. Cook until the onions and bell pepper are tender, about 4 minutes. Add the turkey and garlic, stir again, and cook for another 2 minutes.
Remove from the heat and, using a spoon, create 4 small wells in the hash. Crack an egg into each of the wells. Transfer the skillet to the oven and cook until the eggs are set (or cooked to the desired degree of doneness), about 9 minutes. Remove from the oven, season the eggs with a pinch of salt and pepper, garnish with the parsley, and serve immediately.