Croque Madame Bread Pudding

Name: Croque Madame Bread Pudding
Ingredients: * 5 egg; * 4 cup of milk; * 1/2 cup of 1/2 and 1/2; * 1 tablespoon of dijon mustard; * 1/8 teaspoon of freshly grated nutmeg; * 1/8 teaspoon of cayenne pepper; * 1/2 teaspoon of kosher salt; * 1/4 teaspoon of freshly cracked black pepper; * 1/2 teaspoon of fresh thyme, minced; * 1 1/2 cup of grated monterey jack cheese, divided; * 1 loaf of challah bread,; * 1 pound of ham slice (about 2 large ham slice), diced; * 2 tablespoon of butter; for serving:; * 3 teaspoon of butter, divided; * 8 egg; * salt and pepper, to taste