Preheat oven to 375 F.
Brush the insides of the muffins with 1/4 cup of the butter.
Place onto a baking sheet and bake until lightly golden and starting to get crispy around the edges.
Add 2 tablespoons of the remaining butter to a large over-safe skillet over medium-high heat.
Add the ham and brown on both sides. Remove.
Add remaining butter to the skillet.
Crack eggs into 4 silicon ring molds.
Sprinkle with thyme, salt and pepper.
Reduce heat to low and cook for 2 minutes or until the whites are almost done.
Transfer the skillet to the oven and cook another 2 minutes until the whites are cooked. Remove.
Spread mayonnaise on the muffin halves.
Add the cheese, ham, eggs, a few splashes of hot sauce and the muffin tops.
If desired, place the sandwiches in the oven for 5 minutes to lightly melt the cheese.