Preheat the oven to 375 degrees F. Cut off the top third of the garlic head to expose the cloves. Place onto a piece of foil. Drizzle with some olive oil and sprinkle with salt and pepper. Cover the garlic with the foil and place into the oven for about 1 hour 15 minutes. Remove from the oven and allow to cool to the touch. Remove the garlic cloves from their husks. Reserve until ready to use. Mince 1 teaspoon of the roasted garlic and melt with the butter in a small saucepan. Season with salt. Gently warm until the garlic becomes fragrant. Set aside in the fridge to slightly firm up again. Begin by preparing the bacon for the pig patties. Set a large pot of cold water over high heat and add the bacon. Bring to a medium boil and cook until the bacon is very tender and almost falling apart, 1 1/2 to 2 hours. Drain and set aside to cool. Set a flat cast-iron grill pan over high heat. Divide the bacon into 4 even portions, form each portion into a patty and place on the grill. Press down with the foil-covered brick to squash the bacon together and into a flat patty so it crisps. Cook until golden and crispy, 3 to 4 minutes. Remove and keep warm. Divide the ground beef into 4 equal portions (8 ounces each) and form into 4 tight balls. Sprinkle the hamburger "ball" all over with salt and pepper. To the griddle pan (which still has some of the bacon fat on it) place the hamburger balls on and flatten with a strong, flat metal spatula. Flatten to approximately 1/3-inch thickness and continue to press down with the spatula as it cooks. Cook until the burgers develop a crust on the first side, 1 1/2 minutes. Flip over and cook on the second side to develop a crust on that side, 1 minute. Add 2 ounces Mornay Sauce, if using, to the top of the hamburger and top with a slice of cheese. Place a dome lid over the top of the burger. Pour the chicken broth mixture on to the flat top so the liquid runs under the dome and creates steam to cook the burger and melt the cheese. When the burger is at desired point of doneness and the cheese is completely melted, remove the dome lid and set the burgers aside. Brush the cut sides of the buns lightly with the garlic butter. Toast the buns on the flat top until golden and crisp, 3 to 4 seconds. Flip and toast the outsides so they are also lightly toasted, 3 to 4 seconds. To assemble: Smear the "Donkey" Sauce on the cut side of both bun halves. Layer the base with pickle slices and the hamburger patty. Top with a bacon "pig patty." Top with the onions, tomato and lettuce, and then place the top half of the bun on top. Wrap in waxed parchment paper and serve immediately.
Mornay Sauce (Bechamel with Cheese):
In a medium saucepan, add the butter and set over medium heat. Allow the butter to melt, and then sprinkle with the flour and stir with a whisk. Cook over low heat until the butter and flour combine into a mixture like wet sand. Once combined, stir for 1 to 2 minutes to cook the flour (but without adding color). Pour in the milk slowly as you whisk to combine. Continue stirring until the mixture thickens up, 3 to 4 minutes. Add the salt, nutmeg and cheese and stir until the cheese has melted and the mixture is smooth. Set aside and keep warm until ready to use. Yield: 1 cup.
Mix the mayonnaise, roasted garlic, mustard, Worcestershire, salt and pepper together until smooth. Yield: about 1 cup.