Mr. Crunchy with an Egg (Mrs. Crunchy)

Ingredients

• tablespoons unsalted butter
• ounce all-purpose flour
• teaspoon freshly grated nutmeg
• teaspoon kosher salt
• teaspoon ground white pepper
• fluid ounces whole milk
• teaspoons fresh thyme leaves
• slices sourdough bread
• tablespoons whole-grain mustard
• ounces thinly sliced Black Forest ham, cut into 1-inch wide strips
• ounces Gruyere, grated
• teaspoons herbes de Provence
• tablespoons olive oil
• large eggs
• Kosher salt
• Freshly ground black pepper