For the Duck Bacon:
12 ounces (2 medium) duck breasts
¼ cup kosher salt
¼ cup light brown sugar
½ teaspoon whole black peppercorns
4 thyme sprigs
1 bay leaf
Applewood chips, for smoking
For the Croque Madame:
4 tablespoons unsalted butter, divided
2 tablespoons all-purpose flour
½ cup whole milk
½ cup heavy cream
1 cup grated white cheddar cheese, divided
1 teaspoon Worcestershire sauce
Kosher salt, to taste
8 slices sourdough bread
8 slices Gruyère or Swiss cheese
½ cup grated Parmesan cheese
4 eggs, cooked sunny-side up
1. Make the duck bacon: In a sealable plastic bag, add all of the duck bacon ingredients, except for the wood chips. Seal and shake to combine. Refrigerate for 12 hours to cure. After curing, rinse the duck breasts and dry well with paper towels.
2. Place the duck breasts in a stovetop smoker with the applewood chips set up according to the manufacturer's instructions. Smoke for 1 hour, using the smoker's specifications, then remove the breasts and let cool.
3. Thinly slice the duck breasts lengthwise and place in a cast-iron pan over medium-high heat. Cook until the fat has rendered and the duck bacon is golden, 2 to 3 minutes per side. Set aside.
4. Make the croque madame: In a small saucepan, melt 2 tablespoons of the butter over medium heat. Add the flour and whisk until incorporated, 1 minute. Slowly whisk in the milk and cream until smooth and bring to a simmer. Cook until thickened, 2 to 3 minutes, then stir in half of the grated cheddar and the Worcestershire. Remove from the heat and season with salt.
5. Preheat the broiler. Spread 1 heaping teaspoon of béchamel on both sides of the sliced of bread. Layer 2 slices of Gruyère cheese and a quarter of the duck bacon on the bottom half of the bread, then top with the other bread half and sandwich closed. Repeat for each of the sandwiches.
6. In a large nonstick skillet, heat 1 tablespoon of the butter over medium-high heat. Cook the sandwiches 2 at a time, flipping once, until golden brown, 2 to 3 minutes per side. Transfer the sandwiches to a sheet pan and repeat this process with the remaining butter and sandwiches.
7. Once all of the sandwiches are on the sheet pan, spread the remaining béchamel over each one. Sprinkle with the remaining cheddar and the Parmesan, then place under the broiler. Cook until the cheese is bubbling and golden brown, 3 to 4 minutes.
8. Remove from the oven and divide between plates. Top each croque madame with a sunny-side up egg, then serve.