For the clove-scented ketchup: Whisk together the ketchup and clove in a small bowl. Cover and let sit at room temperature.
For the Home Fries: Heat the butter in a large cast-iron skillet over medium-high heat until it begins to shimmer. Add the potatoes, za'atar, paprika, turmeric and a very generous amount of salt and toss to coat the potatoes. Use a spatula to press and crush the potatoes against the bottom of the pan. Cook until golden brown and crisp on the bottom, about 4 minutes. Turn in large chunks and press again to crisp. Cook until browned on the bottom, another 4 minutes. Taste and add salt as needed.
While the potatoes crisp, heat the vegetable oil in a small skillet over medium heat. Add the chiles and cook, stirring, until warmed and very slightly softened, about 1 minute. Stir in the herbs.
Spoon the chile mixture over the potatoes and drizzle with the lime juice. Serve straight from the pan, with the clove-scented ketchup on the side.