Preheat the oven to 400°F. Pat the tenderloin dry with paper towels, rub it all over with olive oil, and liberally sprinkle all over with salt and pepper. Place it on a baking sheet and roast for 25 to 30 minutes; then, the last 2 minutes, broil it to brown the meat. The internal temperature should measure 130°F on an instant-read thermometer for medium rare. Remove the meat from the oven, tent it with aluminum foil, and let rest for about 20 minutes.

While the tenderloin is roasting, or even in advance, make the Béarnaise sauce. In a small saucepan over medium heat, reduce the vinegar and vermouth with the shallot, salt, pepper, and dry tarragon until only 2 tablespoons remain. Strain the liquid into another saucepan. Over very low heat, whisk the egg yolks into the liquid until thickened, about 2 minutes; you will see whisk marks in the pan. Immediately remove the pan from the heat and, while whisking vigorously, slowly add the hot clear butter by droplets until an emulsion begins to form; then, still whisking, begin adding the clear butter by spoonfuls until all of it has been incorporated, 4 to 5 minutes, leaving behind and discarding the milk solids at the bottom of the measuring cup. Taste the sauce and add salt and pepper as need and, if you like, a little more tarragon vinegar. Hold for serving someplace warm near the stove, but not over direct heat.

To serve, use a sharp knife and carving fork to cut the tenderloin crosswise, across the grain, into generous slices, arranging them on a heated platter. Pour the Béarnaise sauce over the meat and garnish with chopped fresh tarragon or parsley.