Heat 1 tablespoon oil over medium-high heat in a 12-inch non-stick, ovenproof skillet. Add onion; cook 2-3 minutes, stirring often. Add garlic, and cook an additional 2-3 minutes, again stirring often. Add turkey and potatoes and remaining olive oil, continue to cook until golden brown and heated through.
Mix ketchup, mustard, thyme, parsley and water in a small bowl. Stir ketchup mixture into turkey/potatoes. Season with salt and pepper, to taste. Reduce heat to medium, continue to stir for an additional 4-5 minutes.
Make 4 “wells” into hash mixture. Break an egg into a small bowl and transfer egg to the “well.” Repeat until 4 wells are filled with an egg. Transfer skillet to 350-degree, preheated oven. Bake 20-22 minutes, or until egg whites are firmly set and yolks are to your desired degree of doneness.