In a small saucepan, combine the shallots, 1 tablespoon of chopped tarragon, 1 tablespoon of chopped Italian parsley and 1 1/2 teaspoons chopped chives. Add the vinegar and bring to a rapid simmer.
Stir the herb and vinegar mixture and simmer until the liquid is reduced by about half. Set aside to cool.
Add the lemon zest to the butter and mix it well with a fork. Add the remaining chopped tarragon, parsley and chives to the butter mixture.
Place a fine mesh strainer over a small bowl. Pour the simmered vinegar and herbs into the strainer and press on the solids with the back of a wooden spoon to extract as much liquid as possible. Discard the solids.
Add the strained cooled vinegar to the butter and herbs and mix well until the liquid is incorporated in the butter and herbs are evenly distributed.
Tear off a piece of parchment paper about 8" wide. Moud the butter mixture in the center of the parchment. Fold the long end over the mound of butter until the edge touches the parchment beneath. Holding the top piece of parchment down, pull it towards you, so that the butter forms a cylinder. Roll the butter into the parchment and twist the ends to seal. Refrigerate for an hour or until is solid. (Can be made several days in advance).
TO COOK THE VEGGIES & RIBEYE STEAKS:
Heat a large dry cast iron skillet over medium high heat. Add the olive oil, butter and mushrooms. Add half of the kosher salt and sauté, stirring occasionally, until the mushrooms become tender and start to give off some liquid, about 5-8 minutes.
Stir in the sliced shallots and cook for an additional 2-4 minutes or until the shallots are tender and slightly translucent. Transfer to a small dish and top with a small pat of the compound butter.
Pat the steaks dry with a paper towel. Season both sides with the remaining salt and pepper. Set aside.
In the same skillet, turn the heat up to high. When pan is very hot place the steaks in the pan and sear without touching or movingthe meat for 2-3 minutes.
Use a pair of tongs and turn the steaks over. Cook for an additional 2-3 minutes, then check the temperature of the steaks with an instant read thermometer for doneness. I like medium rare and cook them to 130°-135°F (54°-57°C). Transfer the steaks to a serving platter or plate and top with 2 slices of Bearnaise Butter.
Sprinkle with additional chopped herbs and serve with the shallot mushroom mixture.
Note: Leftover Béarnaise butter can be used on chicken, pork, fish and seafood or over steamed vegetables, baked or mashed potatoes, tossed with hot cooked pasta or rice.