Cut the potatoes in 1/2-inch dice. Place in a saucepan, cover with cold water, and add 2 teaspoons salt. Bring to a boil and simmer until crisp tender, about 5 minutes; drain.
Prepare the grill for direct cooking over medium heat (350° to 450°F). Preheat the griddle over direct medium heat for 5 minutes.
Prepare the grill for cooking over medium heat (400°F). Preheat the griddle over direct medium heat for 5 minutes.
Heat the oil and melt the butter on the griddle. Add the onions and sauté until soft, about 3 minutes. Add the peppers and sauté 1 minute. Add the potatoes, ham, 1 teaspoon salt, the thyme, and black pepper and stir to combine. Close the lid and grill until the ham crisps and the potatoes are golden brown in spots, 12 to 15 minutes, stirring occasionally. If the pan dries out while cooking, add 1 tablespoon more oil.
Make four wells in the hash. Crack an egg into a small bowl. Tip the egg into a well. Repeat with the remaining eggs. Close the lid and cook the eggs until set, about 8 minutes.
Season the hash with additional salt and black pepper. Garnish with parsley. Divide the hash between 4 serving plates. Serve immediately.