Preheat the oven to 375 degrees F. Put the tomato halves on a baking sheet, cut-side up, drizzle with oil and season with salt, pepper, and thyme. Roast the tomatoes until soft, about 30 minutes. Remove the tomatoes to a plate. Wipe the baking sheet with paper towels and place the English muffin halves on the baking sheet and toast in the oven until lightly golden brown, about 5 minutes. Top each with a slice of the cheese, return to the oven and toast until the cheese is melted, about 2 minutes. Remove and let sit while you make the eggs.
Heat a nonstick pan over medium heat. Add the ham slices and cook until slightly crisp and warmed through. Remove the ham to a plate. Add a few tablespoons of butter to the pan, whisk 8 large eggs in a bowl, season with salt and pepper and cook over low heat until soft curds form and stir in the chives.
Top each muffin half with a slice of ham. Top the bottom half with some of the eggs and 2 tomato halves. Combine halves to make 4 sandwiches. Serve immediately.
Recipe courtesy of Bobby Flay