Add ground beef to a large bowl and season with salt and pepper. Add in 1 tablespoon of olive oil and mix it into the beef, then let sit at room temperature for about 30 minutes.
While the beef is resting, add mushrooms to a large skillet over low heat. Add 1 tablespoon each of olive oil and butter, then add in mushrooms and stir to coat. Cover and cook for 10-15 minutes until softened and juicy. Once soft, add a sprinkle of salt and pepper, turn off heat and keep covered.
While mushrooms are cooking, whisk together dijon mustard, egg yolk and lemon juice in a large bowl until combined. Slowly stream in olive oil while continuously whisking until the mixture emulsifies and comes together completely. Whisk in whole grain mustard and horseradish, then taste and season with a little salt and pepper as desired. Set aside.
Toss the fresh arugula with the lemon juice and 1/2 tablespoon of olive oil. Sprinkle with a little salt and pepper.
Heat a large skillet over medium-high heat and add 1 tablespoon of butter. Form the beef into 4 equal patties, then add to the skillet and cook until they reach their desired doneness. For medium-well, I cook mine about 5 minutes per side, but it will depend on how thick you like your burgers. In the last 1-2 minutes of cooking, top each burger with a few slices of both cheeses.
To assemble burgers, spread a bit of mustard aioli on the bottom of each bun. Top with the burger, then some mushrooms, some aioli and a handful of arugula.