In a medium pot, add the potatoes and enough water to cover. Bring to a boil, add salt and cook until tender, 12 to 15 minutes. Drain, return the potatoes to the pot and mash with enough milk for your desired consistency. Mash in the cheese; season with salt and pepper.
Heat a cast-iron skillet over medium heat until hot. Drizzle the steaks with EVOO, add to the pan and cook for 8 minutes for medium-rare; season with salt and pepper.
In a small skillet, melt the butter over medium heat. Add the shallot and cook for 2 minutes. Add the tomato paste and paprika and cook for 1 minute. Whisk in the flour for 1 minute, then whisk in the beef broth. Stir in the parsley, capers and Worcestershire sauce. Remove from the heat; season with pepper.
Dress the arugula with the lemon juice, a little EVOO, salt and pepper.
Top the steaks with the gravy and greens. Serve with the potatoes.