Parboil the potatoes in salted water for about 5 minutes, until just tender but still holding their shape. Drain well and let them steam dry for a few minutes in the pot.
While the potatoes cook, pat the chicken hearts dry with paper towels. Heat 1 tablespoon of butter in a large skillet and cook the hearts over medium-high heat for 5–6 minutes, until browned and just cooked through. Season with salt and pepper, then set aside.
Add a little more butter to the pan and fry the potatoes over medium-high heat until golden and crisp. Let them sit undisturbed for a few minutes to brown before stirring. Season and set aside.
If needed, add a bit more butter and sauté the onion and bell pepper until softened. Stir in the thyme (or marjoram) and cook for about 1 minute. Season lightly.
Return the potatoes and chicken hearts to the pan and cook until heated through. Taste and adjust seasoning.
Meanwhile, fry the eggs in a separate skillet until the whites are set and the yolks are still runny.
Serve the hash immediately topped with fried eggs and fresh parsley if desired.