For the croutons: Preheat the oven to 300 degrees F. Heat the oil in a cast-iron skillet over high heat. Add the ham and cook until golden brown. Transfer to a plate lined with paper towels using a slotted spoon. When cool, dice into 1/2-inch pieces.
Turn the heat to medium-high and add the butter and garlic to the skillet. Cook for a few seconds and add the bread. Season with salt and pepper and cook until the croutons are lightly golden brown on all sides. Transfer to a baking sheet and keep warm in the oven.
For the creamy eggs: Melt the butter in a large nonstick saute pan over low heat. Whisk the eggs in a bowl until light and fluffy and sprinkle with salt and pepper. Pour the eggs into the pan and cook slowly over low heat. Stir constantly with a heatproof rubber spatula until soft curds form. Remove the eggs from the heat and stir in the cheese, ham, chives and dill. Transfer to a large platter and top with the croutons.