Preheat the oven to 350 degrees. Heat a cast iron pan on medium heat. Place bacon strips in the pan and cook until all fat is rendered. Pull the bacon out of the pan and place on a paper towel to cool and drain excess fat. Once cooled, chop the bacon into bits. Pour fat out of the pan, reserving about 1/4 to cook the sprouts. Turn heat to low and let the reserved bacon fat heat up. Place Brussels face side down in the hot fat and let them cook until the underside is golden brown. Once golden brown, add 4oz of blue cheese crumbles on top of the sprouts and place in the oven for 10 minutes to finish cooking. Once cooked, pull the pan out of the oven, top with bacon bits and the remaining blue cheese crumbles.
Pour red wine vinegar, shallots, and half of the tarragon into a sauce pot and bring to a rolling boil. Reduce the vinegar by 2/3 the amount. Once reduced, slowly start to whisk in the cubed butter. Only add more butter once the previous cubes are completely melted. After adding most of the butter, add in the egg yolks and whisk vigorously. It is important to control the heat of the sauce as you are whisking, so that the sauce does not get too hot and separate. Finish adding the rest of the butter and whisk until completely emulsified. Pull the sauce off the heat, add in the rest of the tarragon and season with salt and pepper. Set aside.
Toast the brioche bun in a pan on low heat, for a nice golden brown. Heat a separate pan, and add in sliced mushrooms. Saute the mushrooms until soft, season with salt and pepper. Heat a cast iron pan on the stovetop until it is hot to the touch. Season the burger patty and place it in the cast iron, it should sizzle and start to sear the outside of the burger immediately. Cook for about 3 minutes and then flip to the other side, cook for another 3 minutes. Add the Swiss cheese and let it melt as the burger finishes cooking. Pull the burger out of the pan and place it on the bun. Spoon the sauteed mushrooms on top of the burger, and ladle Bearnaise sauce over top. Serve with a portion of Brussels Sprouts and enjoy!