To make the béchamel, place the butter in a small sauce pan and melt over medium heat. Add the flour to the melted butter and whisk to combine. Cook, stirring continuously, for 1 minute. Lower the heat slightly and while whisking constantly, add the milk in a slow, steady stream. Continue to whisk until the sauce has a smooth, silky texture.

Increase the heat back to medium and bring the sauce to a simmer. Whisk in the grated cheddar a few tablespoons at a time, making sure all the cheese has melted before adding more. Once all the cheese has been incorporated, reduce the heat to low and cook, stirring occasionally, till the sauce has thickened. Season to taste with salt.

To assemble the sandwiches, heat a large non-stick pan over medium-high heat. Add the butter to the pan. When the butter starts to foam and sizzle, crack the eggs into the pan and cook to the desired doneness. While the eggs cook, toast the buns. Spread about 1 tablespoon of the Chili Jam over the bottom of each, then add a pinch of greens, a quarter of an avocado, two strips of bacon, and top with the fried egg.

Drizzle with béchamel and more Chili Jam and enjoy.