Generously salt the steak and let it rest at room temperature.
Heat 1 tablespoon of butter in a saucepan over medium heat. Add the shallots. Cook for 1 minute, stirring, then add the vinegar. Cook until the liquid is almost totally reduced. Add the cream and bring to a simmer. Quickly slice the remaining butter into pieces and add them to the saucepan, stirring rapidly with a whisk, until the sauce is smooth. Stir in the blue cheese. Keep warm.
Pat dry the steak with a paper towel. Heat a heavy skillet over medium-high heat. Add the oil, then add the steak and let it sear, untouched, for 3 minutes. Flip the steak and sear the other side for 3 minutes. Flip again and sear for 2 minutes on each side for medium-rare (sear an additional 1 minute per side for medium to medium-well). Let the steak rest on a plate for 5 minutes.
Meanwhile, make the broccoli: Place 1 inch of water, a pinch of salt, and the broccoli in a pot with a lid. Bring to a boil, cover the pot, lower the heat to simmer, and let cook for 5 minutes. Drain.
To serve, slice the steak into strips. Divide the steak and broccoli between 2 dinner plates. Garnish with the blue cheese sauce and enjoy.
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